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Theorem preleq 7318
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 preleq.1 . . . . . . 7  |-  A  e. 
_V
2 preleq.2 . . . . . . 7  |-  B  e. 
_V
3 preleq.3 . . . . . . 7  |-  C  e. 
_V
4 preleq.4 . . . . . . 7  |-  D  e. 
_V
51, 2, 3, 4preq12b 3788 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
65biimpi 186 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
76ord 366 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  ( A  =  D  /\  B  =  C ) ) )
8 en2lp 7317 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
9 eleq12 2345 . . . . . 6  |-  ( ( A  =  D  /\  B  =  C )  ->  ( A  e.  B  <->  D  e.  C ) )
109anbi1d 685 . . . . 5  |-  ( ( A  =  D  /\  B  =  C )  ->  ( ( A  e.  B  /\  C  e.  D )  <->  ( D  e.  C  /\  C  e.  D ) ) )
118, 10mtbiri 294 . . . 4  |-  ( ( A  =  D  /\  B  =  C )  ->  -.  ( A  e.  B  /\  C  e.  D ) )
127, 11syl6 29 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  -.  ( A  e.  B  /\  C  e.  D ) ) )
1312con4d 97 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  e.  B  /\  C  e.  D )  ->  ( A  =  C  /\  B  =  D )
) )
1413impcom 419 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    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {cpr 3641
This theorem is referenced by:  opthreg  7319  dfac2  7757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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