Theorem opcom 4283

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 4270	. 2 |- ( <. A , B >. = <. B , A >. <-> ( A = B /\ B = A ) )
4		eqcom 2198	. . 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 1364   e. wcel 2167   _Vcvv 2763   <.cop 3625

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by: (None)