Theorem nosepnelem 31830

Description: Lemma for nosepne 31831. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

Ref	Expression
nosepnelem	|- ( ( A e. No /\ B e. No /\ A <s B ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem nosepnelem

Step	Hyp	Ref	Expression
1		sltval2 31809	. . 3 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) )
2		fvex 6201	. . . . 5 |- ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) e. _V
3		fvex 6201	. . . . 5 |- ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) e. _V
4	2, 3	brtp 31639	. . . 4 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
5		1n0 7575	. . . . . 6 |- 1o =/= (/)
6		simpl 473	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o )
7		simpr 477	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
8	6, 7	neeq12d 2855	. . . . . 6 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> 1o =/= (/) ) )
9	5, 8	mpbiri 248	. . . . 5 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
10		df-2o 7561	. . . . . . . . . . 11 |- 2o = suc 1o
11		df-1o 7560	. . . . . . . . . . 11 |- 1o = suc (/)
12	10, 11	eqeq12i 2636	. . . . . . . . . 10 |- ( 2o = 1o <-> suc 1o = suc (/) )
13		1on 7567	. . . . . . . . . . 11 |- 1o e. On
14		0elon 5778	. . . . . . . . . . 11 |- (/) e. On
15		suc11 5831	. . . . . . . . . . 11 |- ( ( 1o e. On /\ (/) e. On ) -> ( suc 1o = suc (/) <-> 1o = (/) ) )
16	13, 14, 15	mp2an 708	. . . . . . . . . 10 |- ( suc 1o = suc (/) <-> 1o = (/) )
17	12, 16	bitri 264	. . . . . . . . 9 |- ( 2o = 1o <-> 1o = (/) )
18	17	necon3bii 2846	. . . . . . . 8 |- ( 2o =/= 1o <-> 1o =/= (/) )
19	5, 18	mpbir 221	. . . . . . 7 |- 2o =/= 1o
20	19	necomi 2848	. . . . . 6 |- 1o =/= 2o
21		simpl 473	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o )
22		simpr 477	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o )
23	21, 22	neeq12d 2855	. . . . . 6 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> 1o =/= 2o ) )
24	20, 23	mpbiri 248	. . . . 5 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
25		2on 7568	. . . . . . . . 9 |- 2o e. On
26	25	elexi 3213	. . . . . . . 8 |- 2o e. _V
27	26	prid2 4298	. . . . . . 7 |- 2o e. { 1o , 2o }
28	27	nosgnn0i 31812	. . . . . 6 |- (/) =/= 2o
29		simpl 473	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
30		simpr 477	. . . . . . 7 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o )
31	29, 30	neeq12d 2855	. . . . . 6 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> (/) =/= 2o ) )
32	28, 31	mpbiri 248	. . . . 5 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
33	9, 24, 32	3jaoi 1391	. . . 4 |- ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
34	4, 33	sylbi 207	. . 3 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
35	1, 34	syl6bi 243	. 2 |- ( ( A e. No /\ B e. No ) -> ( A <s B -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) )
36	35	3impia 1261	1 |- ( ( A e. No /\ B e. No /\ A <s B ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   (/)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:  nosepne  31831