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Theorem prneimg 3587
Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
Assertion
Ref Expression
prneimg  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C  /\  B  =/=  D ) )  ->  { A ,  B }  =/=  { C ,  D } ) )

Proof of Theorem prneimg
StepHypRef Expression
1 preq12bg 3586 . . . . 5  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
2 orddi 767 . . . . . 6  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  <->  ( (
( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C ) )  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C )
) ) )
3 simpll 496 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( A  =  C  \/  A  =  D ) )
4 pm1.4 679 . . . . . . . 8  |-  ( ( B  =  D  \/  B  =  C )  ->  ( B  =  C  \/  B  =  D ) )
54ad2antll 475 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( B  =  C  \/  B  =  D ) )
63, 5jca 300 . . . . . 6  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
) )
72, 6sylbi 119 . . . . 5  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  -> 
( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
) )
81, 7syl6bi 161 . . . 4  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
) ) )
9 oranim 841 . . . . . 6  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( -.  A  =  C  /\  -.  A  =  D ) )
10 df-ne 2250 . . . . . . 7  |-  ( A  =/=  C  <->  -.  A  =  C )
11 df-ne 2250 . . . . . . 7  |-  ( A  =/=  D  <->  -.  A  =  D )
1210, 11anbi12i 448 . . . . . 6  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  ( -.  A  =  C  /\  -.  A  =  D
) )
139, 12sylnibr 635 . . . . 5  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( A  =/= 
C  /\  A  =/=  D ) )
14 oranim 841 . . . . . 6  |-  ( ( B  =  C  \/  B  =  D )  ->  -.  ( -.  B  =  C  /\  -.  B  =  D ) )
15 df-ne 2250 . . . . . . 7  |-  ( B  =/=  C  <->  -.  B  =  C )
16 df-ne 2250 . . . . . . 7  |-  ( B  =/=  D  <->  -.  B  =  D )
1715, 16anbi12i 448 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  D )  <->  ( -.  B  =  C  /\  -.  B  =  D
) )
1814, 17sylnibr 635 . . . . 5  |-  ( ( B  =  C  \/  B  =  D )  ->  -.  ( B  =/= 
C  /\  B  =/=  D ) )
1913, 18anim12i 331 . . . 4  |-  ( ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D ) )  -> 
( -.  ( A  =/=  C  /\  A  =/=  D )  /\  -.  ( B  =/=  C  /\  B  =/=  D
) ) )
208, 19syl6 33 . . 3  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  ->  ( -.  ( A  =/=  C  /\  A  =/=  D )  /\  -.  ( B  =/=  C  /\  B  =/=  D
) ) ) )
21 pm4.56 840 . . 3  |-  ( ( -.  ( A  =/= 
C  /\  A  =/=  D )  /\  -.  ( B  =/=  C  /\  B  =/=  D ) )  <->  -.  (
( A  =/=  C  /\  A  =/=  D
)  \/  ( B  =/=  C  /\  B  =/=  D ) ) )
2220, 21syl6ib 159 . 2  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  ->  -.  ( ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C  /\  B  =/=  D ) ) ) )
2322necon2ad 2306 1  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C  /\  B  =/=  D ) )  ->  { A ,  B }  =/=  { C ,  D } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434    =/= wne 2249   {cpr 3418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2612  df-un 2987  df-sn 3423  df-pr 3424
This theorem is referenced by: (None)
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