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Theorem preleq 8023
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 4195 . . . . . 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 8019 . . . . 5  |-  -.  ( D  e.  C  /\  C  e.  D )
9 eleq12 2536 . . . . . 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 1374    e. wcel 1762   _Vcvv 3106   {cpr 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-reg 8007
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-eprel 4784  df-fr 4831
This theorem is referenced by:  opthreg  8024  dfac2  8500
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