Theorem noseponlem 31817

Description: Lemma for nosepon 31818. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)

Assertion

Ref	Expression
noseponlem	|- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. A. x e. On ( A ` x ) = ( B ` x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem noseponlem

Step	Hyp	Ref	Expression
1		nodmon 31803	. . . 4 |- ( A e. No -> dom A e. On )
2	1	3ad2ant1 1082	. . 3 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> dom A e. On )
3		nodmord 31806	. . . . . . 7 |- ( A e. No -> Ord dom A )
4		ordirr 5741	. . . . . . 7 |- ( Ord dom A -> -. dom A e. dom A )
5	3, 4	syl 17	. . . . . 6 |- ( A e. No -> -. dom A e. dom A )
6	5	3ad2ant1 1082	. . . . 5 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. dom A e. dom A )
7		ndmfv 6218	. . . . 5 |- ( -. dom A e. dom A -> ( A ` dom A ) = (/) )
8	6, 7	syl 17	. . . 4 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( A ` dom A ) = (/) )
9		nosgnn0 31811	. . . . . . 7 |- -. (/) e. { 1o , 2o }
10		elno3 31808	. . . . . . . . . . 11 |- ( B e. No <-> ( B : dom B --> { 1o , 2o } /\ dom B e. On ) )
11	10	simplbi 476	. . . . . . . . . 10 |- ( B e. No -> B : dom B --> { 1o , 2o } )
12	11	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> B : dom B --> { 1o , 2o } )
13		simp3 1063	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> dom A e. dom B )
14	12, 13	ffvelrnd 6360	. . . . . . . 8 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( B ` dom A ) e. { 1o , 2o } )
15		eleq1 2689	. . . . . . . 8 |- ( ( B ` dom A ) = (/) -> ( ( B ` dom A ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) )
16	14, 15	syl5ibcom 235	. . . . . . 7 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( ( B ` dom A ) = (/) -> (/) e. { 1o , 2o } ) )
17	9, 16	mtoi 190	. . . . . 6 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. ( B ` dom A ) = (/) )
18	17	neqned 2801	. . . . 5 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( B ` dom A ) =/= (/) )
19	18	necomd 2849	. . . 4 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> (/) =/= ( B ` dom A ) )
20	8, 19	eqnetrd 2861	. . 3 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( A ` dom A ) =/= ( B ` dom A ) )
21		fveq2 6191	. . . . 5 |- ( x = dom A -> ( A ` x ) = ( A ` dom A ) )
22		fveq2 6191	. . . . 5 |- ( x = dom A -> ( B ` x ) = ( B ` dom A ) )
23	21, 22	neeq12d 2855	. . . 4 |- ( x = dom A -> ( ( A ` x ) =/= ( B ` x ) <-> ( A ` dom A ) =/= ( B ` dom A ) ) )
24	23	rspcev 3309	. . 3 |- ( ( dom A e. On /\ ( A ` dom A ) =/= ( B ` dom A ) ) -> E. x e. On ( A ` x ) =/= ( B ` x ) )
25	2, 20, 24	syl2anc 693	. 2 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> E. x e. On ( A ` x ) =/= ( B ` x ) )
26		df-ne 2795	. . . 4 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
27	26	rexbii 3041	. . 3 |- ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> E. x e. On -. ( A ` x ) = ( B ` x ) )
28		rexnal 2995	. . 3 |- ( E. x e. On -. ( A ` x ) = ( B ` x ) <-> -. A. x e. On ( A ` x ) = ( B ` x ) )
29	27, 28	bitri 264	. 2 |- ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> -. A. x e. On ( A ` x ) = ( B ` x ) )
30	25, 29	sylib 208	1 |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. A. x e. On ( A ` x ) = ( B ` x ) )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   {cpr 4179   dom cdm 5114   Ord word 5722   Oncon0 5723   -->wf 5884   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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

This theorem is referenced by:  nosepon  31818