Theorem opeqpr 4255

Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
opeqpr.1	|- A e. _V
opeqpr.2	|- B e. _V
opeqpr.3	|- C e. _V
opeqpr.4	|- D e. _V

Assertion

Ref	Expression
opeqpr	|- ( <. A , B >. = { C , D } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )

Proof of Theorem opeqpr

Step	Hyp	Ref	Expression
1		eqcom 2179	. 2 |- ( <. A , B >. = { C , D } <-> { C , D } = <. A , B >. )
2		opeqpr.1	. . . 4 |- A e. _V
3		opeqpr.2	. . . 4 |- B e. _V
4	2, 3	dfop 3779	. . 3 |- <. A , B >. = { { A } , { A , B } }
5	4	eqeq2i 2188	. 2 |- ( { C , D } = <. A , B >. <-> { C , D } = { { A } , { A , B } } )
6		opeqpr.3	. . 3 |- C e. _V
7		opeqpr.4	. . 3 |- D e. _V
8	2	snex 4187	. . 3 |- { A } e. _V
9		prexg 4213	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
10	2, 3, 9	mp2an 426	. . 3 |- { A , B } e. _V
11	6, 7, 8, 10	preq12b 3772	. 2 |- ( { C , D } = { { A } , { A , B } } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )
12	1, 5, 11	3bitri 206	1 |- ( <. A , B >. = { C , D } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   _Vcvv 2739   {csn 3594   {cpr 3595   <.cop 3597

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603

This theorem is referenced by:  relop  4779