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Theorem opeqpr 4253
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 2179 . 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 3777 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
54eqeq2i 2188 . 2  |-  ( { C ,  D }  =  <. A ,  B >.  <->  { C ,  D }  =  { { A } ,  { A ,  B } } )
6 opeqpr.3 . . 3  |-  C  e. 
_V
7 opeqpr.4 . . 3  |-  D  e. 
_V
82snex 4185 . . 3  |-  { A }  e.  _V
9 prexg 4211 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
102, 3, 9mp2an 426 . . 3  |-  { A ,  B }  e.  _V
116, 7, 8, 10preq12b 3770 . 2  |-  ( { C ,  D }  =  { { A } ,  { A ,  B } }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B }
)  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
121, 5, 113bitri 206 1  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   _Vcvv 2737   {csn 3592   {cpr 3593   <.cop 3595
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601
This theorem is referenced by:  relop  4777
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