Theorem opcom 4228

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 4215	. 2 |- ( <. A , B >. = <. B , A >. <-> ( A = B /\ B = A ) )
4		eqcom 2167	. . 3 |- ( B = A <-> A = B )
5	4	anbi2i 453	. 2 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
6		anidm 394	. 2 |- ( ( A = B /\ A = B ) <-> A = B )
7	3, 5, 6	3bitri 205	1 |- ( <. A , B >. = <. B , A >. <-> A = B )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   <.cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by: (None)