Theorem opeqpr 4238

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 2172	. 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 3764	. . 3 |- <. A , B >. = { { A } , { A , B } }
5	4	eqeq2i 2181	. 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 4171	. . 3 |- { A } e. _V
9		prexg 4196	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
10	2, 3, 9	mp2an 424	. . 3 |- { A , B } e. _V
11	6, 7, 8, 10	preq12b 3757	. 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 703   = wceq 1348   e. wcel 2141   _Vcvv 2730   {csn 3583   {cpr 3584   <.cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  relop  4761