Theorem opcom 4965

Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)

Hypotheses

Ref	Expression
opcom.1	|- A e. _V
opcom.2	|- B e. _V

Assertion

Ref	Expression
opcom	|- ( <. A , B >. = <. B , A >. <-> A = B )

Proof of Theorem opcom

Step	Hyp	Ref	Expression
1		opcom.1	. . 3 |- A e. _V
2		opcom.2	. . 3 |- B e. _V
3	1, 2	opth 4945	. 2 |- ( <. A , B >. = <. B , A >. <-> ( A = B /\ B = A ) )
4		eqcom 2629	. . 3 |- ( B = A <-> A = B )
5	4	anbi2i 730	. 2 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
6		anidm 676	. 2 |- ( ( A = B /\ A = B ) <-> A = B )
7	3, 5, 6	3bitri 286	1 |- ( <. A , B >. = <. B , A >. <-> A = B )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183

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-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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

This theorem is referenced by: (None)