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Theorem preleq 8122
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 4151 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
65biimpi 198 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
76ord 379 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  ( A  =  D  /\  B  =  C ) ) )
8 en2lp 8118 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
9 eleq12 2519 . . . . . 6  |-  ( ( A  =  D  /\  B  =  C )  ->  ( A  e.  B  <->  D  e.  C ) )
109anbi1d 711 . . . . 5  |-  ( ( A  =  D  /\  B  =  C )  ->  ( ( A  e.  B  /\  C  e.  D )  <->  ( D  e.  C  /\  C  e.  D ) ) )
118, 10mtbiri 305 . . . 4  |-  ( ( A  =  D  /\  B  =  C )  ->  -.  ( A  e.  B  /\  C  e.  D ) )
127, 11syl6 34 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =  C  /\  B  =  D )  ->  -.  ( A  e.  B  /\  C  e.  D ) ) )
1312con4d 109 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  e.  B  /\  C  e.  D )  ->  ( A  =  C  /\  B  =  D )
) )
1413impcom 432 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 370    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   {cpr 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-reg 8107
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-eprel 4745  df-fr 4793
This theorem is referenced by:  opthreg  8123  dfac2  8561
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