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Theorem prneimg 3613
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 3612 . . . . 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 769 . . . . . 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 pm1.4 681 . . . . . . . 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 845 . . . . . 6  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( -.  A  =  C  /\  -.  A  =  D ) )
10 df-ne 2256 . . . . . . 7  |-  ( A  =/=  C  <->  -.  A  =  C )
11 df-ne 2256 . . . . . . 7  |-  ( A  =/=  D  <->  -.  A  =  D )
1210, 11anbi12i 448 . . . . . 6  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  ( -.  A  =  C  /\  -.  A  =  D
) )
139, 12sylnibr 637 . . . . 5  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( A  =/= 
C  /\  A  =/=  D ) )
14 oranim 845 . . . . . 6  |-  ( ( B  =  C  \/  B  =  D )  ->  -.  ( -.  B  =  C  /\  -.  B  =  D ) )
15 df-ne 2256 . . . . . . 7  |-  ( B  =/=  C  <->  -.  B  =  C )
16 df-ne 2256 . . . . . . 7  |-  ( B  =/=  D  <->  -.  B  =  D )
1715, 16anbi12i 448 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  D )  <->  ( -.  B  =  C  /\  -.  B  =  D
) )
1814, 17sylnibr 637 . . . . 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 pm4.56 844 . . 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 2312 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 664    = wceq 1289    e. wcel 1438    =/= wne 2255   {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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