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Theorem preleq 7926
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 4148 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
65biimpi 194 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
76ord 377 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  ( A  =  D  /\  B  =  C ) ) )
8 en2lp 7922 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
9 eleq12 2527 . . . . . 6  |-  ( ( A  =  D  /\  B  =  C )  ->  ( A  e.  B  <->  D  e.  C ) )
109anbi1d 704 . . . . 5  |-  ( ( A  =  D  /\  B  =  C )  ->  ( ( A  e.  B  /\  C  e.  D )  <->  ( D  e.  C  /\  C  e.  D ) ) )
118, 10mtbiri 303 . . . 4  |-  ( ( A  =  D  /\  B  =  C )  ->  -.  ( A  e.  B  /\  C  e.  D ) )
127, 11syl6 33 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  -.  ( A  e.  B  /\  C  e.  D ) ) )
1312con4d 105 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  e.  B  /\  C  e.  D )  ->  ( A  =  C  /\  B  =  D )
) )
1413impcom 430 1  |-  ( ( ( A  e.  B  /\  C  e.  D
)  /\  { A ,  B }  =  { C ,  D }
)  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070   {cpr 3979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-reg 7910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-eprel 4732  df-fr 4779
This theorem is referenced by:  opthreg  7927  dfac2  8403
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