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Theorem preleq 4440
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 4439 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
2 eleq12 2182 . . . . . 6  |-  ( ( A  =  D  /\  B  =  C )  ->  ( A  e.  B  <->  D  e.  C ) )
32anbi1d 460 . . . . 5  |-  ( ( A  =  D  /\  B  =  C )  ->  ( ( A  e.  B  /\  C  e.  D )  <->  ( D  e.  C  /\  C  e.  D ) ) )
41, 3mtbiri 649 . . . 4  |-  ( ( A  =  D  /\  B  =  C )  ->  -.  ( A  e.  B  /\  C  e.  D ) )
54con2i 601 . . 3  |-  ( ( A  e.  B  /\  C  e.  D )  ->  -.  ( A  =  D  /\  B  =  C ) )
65adantr 274 . 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 3667 . . . 4  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1211biimpi 119 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1312adantl 275 . 2  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) )
146, 13ecased 1312 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 103    \/ wo 682    = wceq 1316    e. wcel 1465   _Vcvv 2660   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  opthreg  4441
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