Theorem nodense 31842

Description: Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nodense	|- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> E. x e. No ( ( bday ` x ) e. ( bday ` A ) /\ A <s x /\ x <s B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem nodense

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

Step	Hyp	Ref	Expression
1		nodenselem6 31839	. 2 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No )
2		bdayval 31801	. . . . 5 |- ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No -> ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) = dom ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
3	1, 2	syl 17	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) = dom ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
4		dmres 5419	. . . . 5 |- dom ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } i^i dom A )
5		nodenselem5 31838	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) )
6		bdayfo 31828	. . . . . . . . . . 11 |- bday : No -onto-> On
7		fof 6115	. . . . . . . . . . 11 |- ( bday : No -onto-> On -> bday : No --> On )
8	6, 7	ax-mp 5	. . . . . . . . . 10 |- bday : No --> On
9		0elon 5778	. . . . . . . . . 10 |- (/) e. On
10	8, 9	f0cli 6370	. . . . . . . . 9 |- ( bday ` A ) e. On
11	10	onelssi 5836	. . . . . . . 8 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ ( bday ` A ) )
12	5, 11	syl 17	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ ( bday ` A ) )
13		bdayval 31801	. . . . . . . 8 |- ( A e. No -> ( bday ` A ) = dom A )
14	13	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( bday ` A ) = dom A )
15	12, 14	sseqtrd 3641	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ dom A )
16		df-ss 3588	. . . . . 6 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ dom A <-> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } i^i dom A ) = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
17	15, 16	sylib 208	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } i^i dom A ) = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
18	4, 17	syl5eq 2668	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> dom ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
19	3, 18	eqtrd 2656	. . 3 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
20	19, 5	eqeltrd 2701	. 2 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) e. ( bday ` A ) )
21		nodenselem4 31837	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22	21	adantrl 752	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
23		nodenselem8 31841	. . . . . . . . . . . . 13 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B <-> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
24	23	biimpd 219	. . . . . . . . . . . 12 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
25	24	3expia 1267	. . . . . . . . . . 11 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) -> ( A <s B -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) ) )
26	25	imp32 449	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) )
27	26	simpld 475	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o )
28		eqid 2622	. . . . . . . . 9 |- (/) = (/)
29	27, 28	jctir 561	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ (/) = (/) ) )
30	29	3mix1d 1236	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ (/) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ (/) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ (/) = 2o ) ) )
31		fvex 6201	. . . . . . . 8 |- ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
32		0ex 4790	. . . . . . . 8 |- (/) e. _V
33	31, 32	brtp 31639	. . . . . . 7 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } (/) <-> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ (/) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ (/) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ (/) = 2o ) ) )
34	30, 33	sylibr 224	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } (/) )
35	19	fveq2d 6195	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
36		fvnobday 31829	. . . . . . . 8 |- ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) = (/) )
37	1, 36	syl 17	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) = (/) )
38	35, 37	eqtr3d 2658	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
39	34, 38	breqtrrd 4681	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
40		fvres 6207	. . . . . . 7 |- ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( A ` y ) )
41	40	eqcomd 2628	. . . . . 6 |- ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) )
42	41	rgen 2922	. . . . 5 |- A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y )
43	39, 42	jctil 560	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
44		raleq 3138	. . . . . 6 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) <-> A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) ) )
45		fveq2 6191	. . . . . . 7 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` x ) = ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
46		fveq2 6191	. . . . . . 7 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
47	45, 46	breq12d 4666	. . . . . 6 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
48	44, 47	anbi12d 747	. . . . 5 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) ) <-> ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
49	48	rspcev 3309	. . . 4 |- ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) ) )
50	22, 43, 49	syl2anc 693	. . 3 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) ) )
51		simpll 790	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> A e. No )
52		sltval 31800	. . . 4 |- ( ( A e. No /\ ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No ) -> ( A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> E. x e. On ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) ) ) )
53	51, 1, 52	syl2anc 693	. . 3 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> E. x e. On ( A. y e. x ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) ) ) )
54	50, 53	mpbird 247	. 2 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
55	41	adantl 482	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) /\ y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( A ` y ) = ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) )
56		nodenselem7 31840	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A ` y ) = ( B ` y ) ) )
57	56	imp 445	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) /\ y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( A ` y ) = ( B ` y ) )
58	55, 57	eqtr3d 2658	. . . . 5 |- ( ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) /\ y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) )
59	58	ralrimiva 2966	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) )
60	26	simprd 479	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o )
61	60, 28	jctil 560	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( (/) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) )
62	61	3mix3d 1238	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( (/) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( (/) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( (/) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
63		fvex 6201	. . . . . . 7 |- ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
64	32, 63	brtp 31639	. . . . . 6 |- ( (/) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> ( ( (/) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( (/) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( (/) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
65	62, 64	sylibr 224	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> (/) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
66	38, 65	eqbrtrd 4675	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
67		raleq 3138	. . . . . 6 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( A. y e. x ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) <-> A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) ) )
68		fveq2 6191	. . . . . . 7 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( B ` x ) = ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
69	46, 68	breq12d 4666	. . . . . 6 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
70	67, 69	anbi12d 747	. . . . 5 |- ( x = |^| { a e. On | ( A ` a ) =/= ( B ` a ) } -> ( ( A. y e. x ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) <-> ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) ) )
71	70	rspcev 3309	. . . 4 |- ( ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` 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 |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
72	22, 59, 66, 71	syl12anc 1324	. . 3 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> E. x e. On ( A. y e. x ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
73		simplr 792	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> B e. No )
74		sltval 31800	. . . 4 |- ( ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No /\ B e. No ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B <-> E. x e. On ( A. y e. x ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
75	1, 73, 74	syl2anc 693	. . 3 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B <-> E. x e. On ( A. y e. x ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` y ) = ( B ` y ) /\ ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
76	72, 75	mpbird 247	. 2 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B )
77		fveq2 6191	. . . . 5 |- ( x = ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( bday ` x ) = ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
78	77	eleq1d 2686	. . . 4 |- ( x = ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( ( bday ` x ) e. ( bday ` A ) <-> ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) e. ( bday ` A ) ) )
79		breq2 4657	. . . 4 |- ( x = ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( A <s x <-> A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
80		breq1 4656	. . . 4 |- ( x = ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( x <s B <-> ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B ) )
81	78, 79, 80	3anbi123d 1399	. . 3 |- ( x = ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( ( ( bday ` x ) e. ( bday ` A ) /\ A <s x /\ x <s B ) <-> ( ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) e. ( bday ` A ) /\ A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) /\ ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B ) ) )
82	81	rspcev 3309	. 2 |- ( ( ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. No /\ ( ( bday ` ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) e. ( bday ` A ) /\ A <s ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) /\ ( A |` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <s B ) ) -> E. x e. No ( ( bday ` x ) e. ( bday ` A ) /\ A <s x /\ x <s B ) )
83	1, 20, 54, 76, 82	syl13anc 1328	1 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> E. x e. No ( ( bday ` x ) e. ( bday ` A ) /\ A <s x /\ x <s B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   dom cdm 5114   |` cres 5116   Oncon0 5723   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794   bdaycbday 31795

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798

This theorem is referenced by:  nocvxminlem  31893