Theorem opcom 4349

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 4335	. 2 |- ( <. A , B >. = <. B , A >. <-> ( A = B /\ B = A ) )
4		eqcom 2233	. . 3 |- ( B = A <-> A = B )
5	4	anbi2i 457	. 2 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
6		anidm 396	. 2 |- ( ( A = B /\ A = B ) <-> A = B )
7	3, 5, 6	3bitri 206	1 |- ( <. A , B >. = <. B , A >. <-> A = B )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by: (None)