Theorem opeqpr 4749

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
opeqpr.2
opeqpr.3
opeqpr.4

Assertion

Ref	Expression
opeqpr

Proof of Theorem opeqpr

Step	Hyp	Ref	Expression
1		eqcom 2466	. 2
2		opeqpr.1	. . . 4
3		opeqpr.2	. . . 4
4	2, 3	dfop 4216	. . 3
5	4	eqeq2i 2475	. 2
6		opeqpr.3	. . 3
7		opeqpr.4	. . 3
8		snex 4693	. . 3
9		prex 4694	. . 3
10	6, 7, 8, 9	preq12b 4206	. 2
11	1, 5, 10	3bitri 271	1

Syntax hints:  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  {csn 4029  {cpr 4031  <.cop 4035

This theorem is referenced by:  relop  5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036