Theorem opeqpr 4389

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 2240	. 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 3898	. . 3 |- <. A , B >. = { { A } , { A , B } }
5	4	eqeq2i 2249	. 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 4317	. . 3 |- { A } e. _V
9		prexg 4344	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
10	2, 3, 9	mp2an 430	. . 3 |- { A , B } e. _V
11	6, 7, 8, 10	preq12b 3890	. 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 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   {cpr 3706   <.cop 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  relop  4925