Theorem opeqpr 4968

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

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 2629	. 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 4401	. . 3 |- <. A , B >. = { { A } , { A , B } }
5	4	eqeq2i 2634	. 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		snex 4908	. . 3 |- { A } e. _V
9		prex 4909	. . 3 |- { A , B } e. _V
10	6, 7, 8, 9	preq12b 4382	. 2 |- ( { C , D } = { { A } , { A , B } } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )
11	1, 5, 10	3bitri 286	1 |- ( <. A , B >. = { C , D } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  propeqop  4970  relop  5272