Theorem funsndifnop 39022

Description: A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.)

Hypotheses

Ref	Expression
funsndifnop.a	|- A e. V
funsndifnop.b	|- B e. W
funsndifnop.g	|- G = { <. A , B >. }

Assertion

Ref	Expression
funsndifnop	|- ( A =/= B -> -. G e. ( _V X. _V ) )

Proof of Theorem funsndifnop

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

Step	Hyp	Ref	Expression
1		elvv 4893	. . 3 |- ( G e. ( _V X. _V ) <-> E. x E. y G = <. x , y >. )
2		funsndifnop.g	. . . . . 6 |- G = { <. A , B >. }
3		funsndifnop.a	. . . . . . . . 9 |- A e. V
4	3	elexi 3055	. . . . . . . 8 |- A e. _V
5		funsndifnop.b	. . . . . . . . 9 |- B e. W
6	5	elexi 3055	. . . . . . . 8 |- B e. _V
7	4, 6	funsn 5630	. . . . . . 7 |- Fun { <. A , B >. }
8		funeq 5601	. . . . . . 7 |- ( G = { <. A , B >. } -> ( Fun G <-> Fun { <. A , B >. } ) )
9	7, 8	mpbiri 237	. . . . . 6 |- ( G = { <. A , B >. } -> Fun G )
10	2, 9	ax-mp 5	. . . . 5 |- Fun G
11		funeq 5601	. . . . . . 7 |- ( G = <. x , y >. -> ( Fun G <-> Fun <. x , y >. ) )
12		vex 3048	. . . . . . . 8 |- x e. _V
13		vex 3048	. . . . . . . 8 |- y e. _V
14	12, 13	funop 39019	. . . . . . 7 |- ( Fun <. x , y >. <-> E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) )
15	11, 14	syl6bb 265	. . . . . 6 |- ( G = <. x , y >. -> ( Fun G <-> E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) ) )
16		eqeq2 2462	. . . . . . . . . . 11 |- ( <. x , y >. = { <. a , a >. } -> ( G = <. x , y >. <-> G = { <. a , a >. } ) )
17		eqeq1 2455	. . . . . . . . . . . . 13 |- ( G = { <. A , B >. } -> ( G = { <. a , a >. } <-> { <. A , B >. } = { <. a , a >. } ) )
18		opex 4664	. . . . . . . . . . . . . . 15 |- <. A , B >. e. _V
19	18	sneqr 4139	. . . . . . . . . . . . . 14 |- ( { <. A , B >. } = { <. a , a >. } -> <. A , B >. = <. a , a >. )
20	4, 6	opth 4676	. . . . . . . . . . . . . . 15 |- ( <. A , B >. = <. a , a >. <-> ( A = a /\ B = a ) )
21		eqtr3 2472	. . . . . . . . . . . . . . . 16 |- ( ( A = a /\ B = a ) -> A = B )
22	21	a1d 26	. . . . . . . . . . . . . . 15 |- ( ( A = a /\ B = a ) -> ( x = { a } -> A = B ) )
23	20, 22	sylbi 199	. . . . . . . . . . . . . 14 |- ( <. A , B >. = <. a , a >. -> ( x = { a } -> A = B ) )
24	19, 23	syl 17	. . . . . . . . . . . . 13 |- ( { <. A , B >. } = { <. a , a >. } -> ( x = { a } -> A = B ) )
25	17, 24	syl6bi 232	. . . . . . . . . . . 12 |- ( G = { <. A , B >. } -> ( G = { <. a , a >. } -> ( x = { a } -> A = B ) ) )
26	2, 25	ax-mp 5	. . . . . . . . . . 11 |- ( G = { <. a , a >. } -> ( x = { a } -> A = B ) )
27	16, 26	syl6bi 232	. . . . . . . . . 10 |- ( <. x , y >. = { <. a , a >. } -> ( G = <. x , y >. -> ( x = { a } -> A = B ) ) )
28	27	com23 81	. . . . . . . . 9 |- ( <. x , y >. = { <. a , a >. } -> ( x = { a } -> ( G = <. x , y >. -> A = B ) ) )
29	28	impcom 432	. . . . . . . 8 |- ( ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> ( G = <. x , y >. -> A = B ) )
30	29	exlimiv 1776	. . . . . . 7 |- ( E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> ( G = <. x , y >. -> A = B ) )
31	30	com12 32	. . . . . 6 |- ( G = <. x , y >. -> ( E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> A = B ) )
32	15, 31	sylbid 219	. . . . 5 |- ( G = <. x , y >. -> ( Fun G -> A = B ) )
33	10, 32	mpi 20	. . . 4 |- ( G = <. x , y >. -> A = B )
34	33	exlimivv 1778	. . 3 |- ( E. x E. y G = <. x , y >. -> A = B )
35	1, 34	sylbi 199	. 2 |- ( G e. ( _V X. _V ) -> A = B )
36	35	necon3ai 2649	1 |- ( A =/= B -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   _Vcvv 3045   {csn 3968   <.cop 3974   X. cxp 4832   Fun wfun 5576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590

This theorem is referenced by:  funsneqopb  39025  snstrvtxval  39137  snstriedgval  39138