Theorem funsneqopb 6419

Description: A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.)

Hypotheses

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

Assertion

Ref	Expression
funsneqopb	|- ( A = B <-> G e. ( _V X. _V ) )

Proof of Theorem funsneqopb

Step	Hyp	Ref	Expression
1		funsndifnop.a	. . 3 |- A e. _V
2		funsndifnop.b	. . 3 |- B e. _V
3		funsndifnop.g	. . 3 |- G = { <. A , B >. }
4	1, 2, 3	funsneqop 6418	. 2 |- ( A = B -> G e. ( _V X. _V ) )
5	1, 2, 3	funsndifnop 6416	. . 3 |- ( A =/= B -> -. G e. ( _V X. _V ) )
6	5	necon4ai 2825	. 2 |- ( G e. ( _V X. _V ) -> A = B )
7	4, 6	impbii 199	1 |- ( A = B <-> G e. ( _V X. _V ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   X. cxp 5112

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: (None)