Theorem sltval2 31809

Description: Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)

Assertion

Ref	Expression
sltval2	|- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )

Distinct variable groups:   A, a   B, a

Proof of Theorem sltval2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sltval 31800	. 2 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
2		fvex 6201	. . . . . . . . . . . . 13 |- ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
3		fvex 6201	. . . . . . . . . . . . 13 |- ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
4	2, 3	brtp 31639	. . . . . . . . . . . 12 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
5		1n0 7575	. . . . . . . . . . . . . . . . 17 |- 1o =/= (/)
6	5	neii 2796	. . . . . . . . . . . . . . . 16 |- -. 1o = (/)
7		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) <-> 1o = (/) ) )
8	6, 7	mtbiri 317	. . . . . . . . . . . . . . 15 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
9		fvprc 6185	. . . . . . . . . . . . . . 15 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
10	8, 9	nsyl2 142	. . . . . . . . . . . . . 14 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
11	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
12	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
13		2on0 7569	. . . . . . . . . . . . . . . . 17 |- 2o =/= (/)
14	13	neii 2796	. . . . . . . . . . . . . . . 16 |- -. 2o = (/)
15		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) <-> 2o = (/) ) )
16	14, 15	mtbiri 317	. . . . . . . . . . . . . . 15 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
17		fvprc 6185	. . . . . . . . . . . . . . 15 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
18	16, 17	nsyl2 142	. . . . . . . . . . . . . 14 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
19	18	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
20	11, 12, 19	3jaoi 1391	. . . . . . . . . . . 12 |- ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
21	4, 20	sylbi 207	. . . . . . . . . . 11 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
22		onintrab 7001	. . . . . . . . . . 11 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V <-> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
23	21, 22	sylib 208	. . . . . . . . . 10 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
24	23	adantl 482	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
25		onelon 5748	. . . . . . . . . . . . . 14 |- ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> y e. On )
26	25	expcom 451	. . . . . . . . . . . . 13 |- ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On -> y e. On ) )
27	24, 26	syl5 34	. . . . . . . . . . . 12 |- ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> y e. On ) )
28		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( a = y -> ( A ` a ) = ( A ` y ) )
29		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( a = y -> ( B ` a ) = ( B ` y ) )
30	28, 29	neeq12d 2855	. . . . . . . . . . . . . 14 |- ( a = y -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` y ) =/= ( B ` y ) ) )
31	30	onnminsb 7004	. . . . . . . . . . . . 13 |- ( y e. On -> ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> -. ( A ` y ) =/= ( B ` y ) ) )
32	31	com12 32	. . . . . . . . . . . 12 |- ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( y e. On -> -. ( A ` y ) =/= ( B ` y ) ) )
33	27, 32	syldc 48	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> -. ( A ` y ) =/= ( B ` y ) ) )
34		df-ne 2795	. . . . . . . . . . . 12 |- ( ( A ` y ) =/= ( B ` y ) <-> -. ( A ` y ) = ( B ` y ) )
35	34	con2bii 347	. . . . . . . . . . 11 |- ( ( A ` y ) = ( B ` y ) <-> -. ( A ` y ) =/= ( B ` y ) )
36	33, 35	syl6ibr 242	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` y ) = ( B ` y ) ) )
37	36	ralrimiv 2965	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) )
38	24, 37	jca 554	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) ) )
39	38	ex 450	. . . . . . 7 |- ( ( A e. No /\ B e. No ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) ) ) )
40	39	impac 651	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
41		anass 681	. . . . . 6 |- ( ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) <-> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
42	40, 41	sylib 208	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
43		raleq 3138	. . . . . . 7 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A. y e. x ( A ` y ) = ( B ` y ) <-> A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) ) )
44		fveq2 6191	. . . . . . . 8 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` x ) = ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
45		fveq2 6191	. . . . . . . 8 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( B ` x ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
46	44, 45	breq12d 4666	. . . . . . 7 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
47	43, 46	anbi12d 747	. . . . . 6 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) <-> ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
48	47	rspcev 3309	. . . . 5 |- ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
49	42, 48	syl 17	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
50	49	ex 450	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
51		eqeq12 2635	. . . . . . . . . . . . . 14 |- ( ( ( A ` x ) = 1o /\ ( B ` x ) = (/) ) -> ( ( A ` x ) = ( B ` x ) <-> 1o = (/) ) )
52	6, 51	mtbiri 317	. . . . . . . . . . . . 13 |- ( ( ( A ` x ) = 1o /\ ( B ` x ) = (/) ) -> -. ( A ` x ) = ( B ` x ) )
53		1on 7567	. . . . . . . . . . . . . . . . 17 |- 1o e. On
54		0elon 5778	. . . . . . . . . . . . . . . . 17 |- (/) e. On
55		suc11 5831	. . . . . . . . . . . . . . . . . 18 |- ( ( 1o e. On /\ (/) e. On ) -> ( suc 1o = suc (/) <-> 1o = (/) ) )
56	55	necon3bid 2838	. . . . . . . . . . . . . . . . 17 |- ( ( 1o e. On /\ (/) e. On ) -> ( suc 1o =/= suc (/) <-> 1o =/= (/) ) )
57	53, 54, 56	mp2an 708	. . . . . . . . . . . . . . . 16 |- ( suc 1o =/= suc (/) <-> 1o =/= (/) )
58	5, 57	mpbir 221	. . . . . . . . . . . . . . 15 |- suc 1o =/= suc (/)
59		df-2o 7561	. . . . . . . . . . . . . . . 16 |- 2o = suc 1o
60		df-1o 7560	. . . . . . . . . . . . . . . 16 |- 1o = suc (/)
61	59, 60	eqeq12i 2636	. . . . . . . . . . . . . . 15 |- ( 2o = 1o <-> suc 1o = suc (/) )
62	58, 61	nemtbir 2889	. . . . . . . . . . . . . 14 |- -. 2o = 1o
63		eqeq12 2635	. . . . . . . . . . . . . . 15 |- ( ( ( A ` x ) = 1o /\ ( B ` x ) = 2o ) -> ( ( A ` x ) = ( B ` x ) <-> 1o = 2o ) )
64		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( 1o = 2o <-> 2o = 1o )
65	63, 64	syl6bb 276	. . . . . . . . . . . . . 14 |- ( ( ( A ` x ) = 1o /\ ( B ` x ) = 2o ) -> ( ( A ` x ) = ( B ` x ) <-> 2o = 1o ) )
66	62, 65	mtbiri 317	. . . . . . . . . . . . 13 |- ( ( ( A ` x ) = 1o /\ ( B ` x ) = 2o ) -> -. ( A ` x ) = ( B ` x ) )
67	13	nesymi 2851	. . . . . . . . . . . . . 14 |- -. (/) = 2o
68		eqeq12 2635	. . . . . . . . . . . . . 14 |- ( ( ( A ` x ) = (/) /\ ( B ` x ) = 2o ) -> ( ( A ` x ) = ( B ` x ) <-> (/) = 2o ) )
69	67, 68	mtbiri 317	. . . . . . . . . . . . 13 |- ( ( ( A ` x ) = (/) /\ ( B ` x ) = 2o ) -> -. ( A ` x ) = ( B ` x ) )
70	52, 66, 69	3jaoi 1391	. . . . . . . . . . . 12 |- ( ( ( ( A ` x ) = 1o /\ ( B ` x ) = (/) ) \/ ( ( A ` x ) = 1o /\ ( B ` x ) = 2o ) \/ ( ( A ` x ) = (/) /\ ( B ` x ) = 2o ) ) -> -. ( A ` x ) = ( B ` x ) )
71		fvex 6201	. . . . . . . . . . . . 13 |- ( A ` x ) e. _V
72		fvex 6201	. . . . . . . . . . . . 13 |- ( B ` x ) e. _V
73	71, 72	brtp 31639	. . . . . . . . . . . 12 |- ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( ( ( A ` x ) = 1o /\ ( B ` x ) = (/) ) \/ ( ( A ` x ) = 1o /\ ( B ` x ) = 2o ) \/ ( ( A ` x ) = (/) /\ ( B ` x ) = 2o ) ) )
74		df-ne 2795	. . . . . . . . . . . 12 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
75	70, 73, 74	3imtr4i 281	. . . . . . . . . . 11 |- ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) -> ( A ` x ) =/= ( B ` x ) )
76		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
77		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
78	76, 77	neeq12d 2855	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
79	78	elrab 3363	. . . . . . . . . . . . . 14 |- ( x e. { a e. On | ( A ` a ) =/= ( B ` a ) } <-> ( x e. On /\ ( A ` x ) =/= ( B ` x ) ) )
80	79	biimpri 218	. . . . . . . . . . . . 13 |- ( ( x e. On /\ ( A ` x ) =/= ( B ` x ) ) -> x e. { a e. On | ( A ` a ) =/= ( B ` a ) } )
81	80	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) =/= ( B ` x ) ) -> x e. { a e. On | ( A ` a ) =/= ( B ` a ) } )
82		ssrab2 3687	. . . . . . . . . . . . . . . . . 18 |- { a e. On | ( A ` a ) =/= ( B ` a ) } C_ On
83		ne0i 3921	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { a e. On | ( A ` a ) =/= ( B ` a ) } -> { a e. On | ( A ` a ) =/= ( B ` a ) } =/= (/) )
84	83	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> { a e. On | ( A ` a ) =/= ( B ` a ) } =/= (/) )
85		onint 6995	. . . . . . . . . . . . . . . . . 18 |- ( ( { a e. On | ( A ` a ) =/= ( B ` a ) } C_ On /\ { a e. On | ( A ` a ) =/= ( B ` a ) } =/= (/) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. { a e. On | ( A ` a ) =/= ( B ` a ) } )
86	82, 84, 85	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. { a e. On | ( A ` a ) =/= ( B ` a ) } )
87		nfrab1 3122	. . . . . . . . . . . . . . . . . . . 20 |- F/_ a { a e. On | ( A ` a ) =/= ( B ` a ) }
88	87	nfint 4486	. . . . . . . . . . . . . . . . . . 19 |- F/_ a |^| { a e. On | ( A ` a ) =/= ( B ` a ) }
89		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 |- F/_ a On
90		nfcv 2764	. . . . . . . . . . . . . . . . . . . . 21 |- F/_ a A
91	90, 88	nffv 6198	. . . . . . . . . . . . . . . . . . . 20 |- F/_ a ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
92		nfcv 2764	. . . . . . . . . . . . . . . . . . . . 21 |- F/_ a B
93	92, 88	nffv 6198	. . . . . . . . . . . . . . . . . . . 20 |- F/_ a ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
94	91, 93	nfne 2894	. . . . . . . . . . . . . . . . . . 19 |- F/ a ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
95		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( a = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` a ) = ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
96		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( a = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( B ` a ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
97	95, 96	neeq12d 2855	. . . . . . . . . . . . . . . . . . 19 |- ( a = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
98	88, 89, 94, 97	elrabf 3360	. . . . . . . . . . . . . . . . . 18 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. { a e. On | ( A ` a ) =/= ( B ` a ) } <-> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
99	98	simprbi 480	. . . . . . . . . . . . . . . . 17 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
100	86, 99	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
101		df-ne 2795	. . . . . . . . . . . . . . . 16 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) =/= ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
102	100, 101	sylib 208	. . . . . . . . . . . . . . 15 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
103		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( y = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` y ) = ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
104		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( y = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( B ` y ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
105	103, 104	eqeq12d 2637	. . . . . . . . . . . . . . . . 17 |- ( y = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A ` y ) = ( B ` y ) <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
106	105	rspccv 3306	. . . . . . . . . . . . . . . 16 |- ( A. y e. x ( A ` y ) = ( B ` y ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. x -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
107	106	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. x -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
108	102, 107	mtod 189	. . . . . . . . . . . . . 14 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. x )
109		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> x e. On )
110		oninton 7000	. . . . . . . . . . . . . . . . 17 |- ( ( { a e. On | ( A ` a ) =/= ( B ` a ) } C_ On /\ { a e. On | ( A ` a ) =/= ( B ` a ) } =/= (/) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
111	82, 83, 110	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( x e. { a e. On | ( A ` a ) =/= ( B ` a ) } -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
112	111	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
113		ontri1 5757	. . . . . . . . . . . . . . 15 |- ( ( x e. On /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On ) -> ( x C_ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } <-> -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. x ) )
114	109, 112, 113	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( x C_ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } <-> -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. x ) )
115	108, 114	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> x C_ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
116		intss1 4492	. . . . . . . . . . . . . 14 |- ( x e. { a e. On | ( A ` a ) =/= ( B ` a ) } -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ x )
117	116	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ x )
118	115, 117	eqssd 3620	. . . . . . . . . . . 12 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ x e. { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
119	81, 118	syldan 487	. . . . . . . . . . 11 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) =/= ( B ` x ) ) -> x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
120	75, 119	sylan2 491	. . . . . . . . . 10 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
121	120	fveq2d 6195	. . . . . . . . 9 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( A ` x ) = ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
122	120	fveq2d 6195	. . . . . . . . 9 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( B ` x ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
123	121, 122	breq12d 4666	. . . . . . . 8 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
124	123	biimpd 219	. . . . . . 7 |- ( ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
125	124	ex 450	. . . . . 6 |- ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
126	125	pm2.43d 53	. . . . 5 |- ( ( x e. On /\ A. y e. x ( A ` y ) = ( B ` y ) ) -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
127	126	expimpd 629	. . . 4 |- ( x e. On -> ( ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
128	127	rexlimiv 3027	. . 3 |- ( E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
129	50, 128	impbid1 215	. 2 |- ( ( A e. No /\ B e. No ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
130	1, 129	bitr4d 271	1 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   Oncon0 5723   suc csuc 5725   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fv 5896  df-1o 7560  df-2o 7561  df-slt 31797

This theorem is referenced by:  sltintdifex  31814  sltres  31815  noextendlt  31822  noextendgt  31823  nosepnelem  31830  nosep1o  31832  nosepdmlem  31833  nodenselem8  31841  nosupbnd2lem1  31861