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Theorem preleq 4371
Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1  |-  A  e. 
_V
preleq.2  |-  B  e. 
_V
preleq.3  |-  C  e. 
_V
preleq.4  |-  D  e. 
_V
Assertion
Ref Expression
preleq  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem preleq
StepHypRef Expression
1 en2lp 4370 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
2 eleq12 2152 . . . . . 6  |-  ( ( A  =  D  /\  B  =  C )  ->  ( A  e.  B  <->  D  e.  C ) )
32anbi1d 453 . . . . 5  |-  ( ( A  =  D  /\  B  =  C )  ->  ( ( A  e.  B  /\  C  e.  D )  <->  ( D  e.  C  /\  C  e.  D ) ) )
41, 3mtbiri 635 . . . 4  |-  ( ( A  =  D  /\  B  =  C )  ->  -.  ( A  e.  B  /\  C  e.  D ) )
54con2i 592 . . 3  |-  ( ( A  e.  B  /\  C  e.  D )  ->  -.  ( A  =  D  /\  B  =  C ) )
65adantr 270 . 2  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  -.  ( A  =  D  /\  B  =  C )
)
7 preleq.1 . . . . 5  |-  A  e. 
_V
8 preleq.2 . . . . 5  |-  B  e. 
_V
9 preleq.3 . . . . 5  |-  C  e. 
_V
10 preleq.4 . . . . 5  |-  D  e. 
_V
117, 8, 9, 10preq12b 3614 . . . 4  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1211biimpi 118 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1312adantl 271 . 2  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) )
146, 13ecased 1285 1  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   _Vcvv 2619   {cpr 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 3001  df-un 3003  df-sn 3452  df-pr 3453
This theorem is referenced by:  opthreg  4372
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