Theorem opeqpr 4113

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 2102	. 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 3651	. . 3 |- <. A , B >. = { { A } , { A , B } }
5	4	eqeq2i 2110	. 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 4049	. . 3 |- { A } e. _V
9		prexg 4071	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
10	2, 3, 9	mp2an 420	. . 3 |- { A , B } e. _V
11	6, 7, 8, 10	preq12b 3644	. 2 |- ( { C , D } = { { A } , { A , B } } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )
12	1, 5, 11	3bitri 205	1 |- ( <. A , B >. = { C , D } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   _Vcvv 2641   {csn 3474   {cpr 3475   <.cop 3477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483

This theorem is referenced by:  relop  4627