Theorem funsndifnop 6416

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. _V
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 5177	. . 3 |- ( G e. ( _V X. _V ) <-> E. x E. y G = <. x , y >. )
2		funsndifnop.g	. . . . . 6 |- G = { <. A , B >. }
3		funsndifnop.a	. . . . . . . 8 |- A e. _V
4		funsndifnop.b	. . . . . . . 8 |- B e. _V
5	3, 4	funsn 5939	. . . . . . 7 |- Fun { <. A , B >. }
6		funeq 5908	. . . . . . 7 |- ( G = { <. A , B >. } -> ( Fun G <-> Fun { <. A , B >. } ) )
7	5, 6	mpbiri 248	. . . . . 6 |- ( G = { <. A , B >. } -> Fun G )
8	2, 7	ax-mp 5	. . . . 5 |- Fun G
9		funeq 5908	. . . . . . 7 |- ( G = <. x , y >. -> ( Fun G <-> Fun <. x , y >. ) )
10		vex 3203	. . . . . . . 8 |- x e. _V
11		vex 3203	. . . . . . . 8 |- y e. _V
12	10, 11	funop 6414	. . . . . . 7 |- ( Fun <. x , y >. <-> E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) )
13	9, 12	syl6bb 276	. . . . . 6 |- ( G = <. x , y >. -> ( Fun G <-> E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) ) )
14		eqeq2 2633	. . . . . . . . . . 11 |- ( <. x , y >. = { <. a , a >. } -> ( G = <. x , y >. <-> G = { <. a , a >. } ) )
15		eqeq1 2626	. . . . . . . . . . . . 13 |- ( G = { <. A , B >. } -> ( G = { <. a , a >. } <-> { <. A , B >. } = { <. a , a >. } ) )
16		opex 4932	. . . . . . . . . . . . . . 15 |- <. A , B >. e. _V
17	16	sneqr 4371	. . . . . . . . . . . . . 14 |- ( { <. A , B >. } = { <. a , a >. } -> <. A , B >. = <. a , a >. )
18	3, 4	opth 4945	. . . . . . . . . . . . . . 15 |- ( <. A , B >. = <. a , a >. <-> ( A = a /\ B = a ) )
19		eqtr3 2643	. . . . . . . . . . . . . . . 16 |- ( ( A = a /\ B = a ) -> A = B )
20	19	a1d 25	. . . . . . . . . . . . . . 15 |- ( ( A = a /\ B = a ) -> ( x = { a } -> A = B ) )
21	18, 20	sylbi 207	. . . . . . . . . . . . . 14 |- ( <. A , B >. = <. a , a >. -> ( x = { a } -> A = B ) )
22	17, 21	syl 17	. . . . . . . . . . . . 13 |- ( { <. A , B >. } = { <. a , a >. } -> ( x = { a } -> A = B ) )
23	15, 22	syl6bi 243	. . . . . . . . . . . 12 |- ( G = { <. A , B >. } -> ( G = { <. a , a >. } -> ( x = { a } -> A = B ) ) )
24	2, 23	ax-mp 5	. . . . . . . . . . 11 |- ( G = { <. a , a >. } -> ( x = { a } -> A = B ) )
25	14, 24	syl6bi 243	. . . . . . . . . 10 |- ( <. x , y >. = { <. a , a >. } -> ( G = <. x , y >. -> ( x = { a } -> A = B ) ) )
26	25	com23 86	. . . . . . . . 9 |- ( <. x , y >. = { <. a , a >. } -> ( x = { a } -> ( G = <. x , y >. -> A = B ) ) )
27	26	impcom 446	. . . . . . . 8 |- ( ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> ( G = <. x , y >. -> A = B ) )
28	27	exlimiv 1858	. . . . . . 7 |- ( E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> ( G = <. x , y >. -> A = B ) )
29	28	com12 32	. . . . . 6 |- ( G = <. x , y >. -> ( E. a ( x = { a } /\ <. x , y >. = { <. a , a >. } ) -> A = B ) )
30	13, 29	sylbid 230	. . . . 5 |- ( G = <. x , y >. -> ( Fun G -> A = B ) )
31	8, 30	mpi 20	. . . 4 |- ( G = <. x , y >. -> A = B )
32	31	exlimivv 1860	. . 3 |- ( E. x E. y G = <. x , y >. -> A = B )
33	1, 32	sylbi 207	. 2 |- ( G e. ( _V X. _V ) -> A = B )
34	33	necon3ai 2819	1 |- ( A =/= B -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {csn 4177   <.cop 4183   X. cxp 5112   Fun wfun 5882

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  funsneqopb  6419  snstrvtxval  25929  snstriedgval  25930