Theorem opcom 4005

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 3992	. 2 |- ( <. A , B >. = <. B , A >. <-> ( A = B /\ B = A ) )
4		eqcom 2083	. . 3 |- ( B = A <-> A = B )
5	4	anbi2i 444	. 2 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
6		anidm 388	. 2 |- ( ( A = B /\ A = B ) <-> A = B )
7	3, 5, 6	3bitri 204	1 |- ( <. A , B >. = <. B , A >. <-> A = B )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by: (None)