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Theorem opeqpr 4445
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
StepHypRef Expression
1 eqcom 2437 . 2  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  { C ,  D }  =  <. A ,  B >. )
2 opeqpr.1 . . . 4  |-  A  e. 
_V
3 opeqpr.2 . . . 4  |-  B  e. 
_V
42, 3dfop 3975 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
54eqeq2i 2445 . 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 4397 . . 3  |-  { A }  e.  _V
9 prex 4398 . . 3  |-  { A ,  B }  e.  _V
106, 7, 8, 9preq12b 3966 . 2  |-  ( { C ,  D }  =  { { A } ,  { A ,  B } }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B }
)  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
111, 5, 103bitri 263 1  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   {cpr 3807   <.cop 3809
This theorem is referenced by:  relop  5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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