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Theorem prneimg 3701
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 3700 . . . . 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 809 . . . . . 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 518 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( A  =  C  \/  A  =  D ) )
4 pm1.4 716 . . . . . . . 8  |-  ( ( B  =  D  \/  B  =  C )  ->  ( B  =  C  \/  B  =  D ) )
54ad2antll 482 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( B  =  C  \/  B  =  D ) )
63, 5jca 304 . . . . . 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 120 . . . . 5  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  -> 
( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
) )
81, 7syl6bi 162 . . . 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 770 . . . . . 6  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( -.  A  =  C  /\  -.  A  =  D ) )
10 df-ne 2309 . . . . . . 7  |-  ( A  =/=  C  <->  -.  A  =  C )
11 df-ne 2309 . . . . . . 7  |-  ( A  =/=  D  <->  -.  A  =  D )
1210, 11anbi12i 455 . . . . . 6  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  ( -.  A  =  C  /\  -.  A  =  D
) )
139, 12sylnibr 666 . . . . 5  |-  ( ( A  =  C  \/  A  =  D )  ->  -.  ( A  =/= 
C  /\  A  =/=  D ) )
14 oranim 770 . . . . . 6  |-  ( ( B  =  C  \/  B  =  D )  ->  -.  ( -.  B  =  C  /\  -.  B  =  D ) )
15 df-ne 2309 . . . . . . 7  |-  ( B  =/=  C  <->  -.  B  =  C )
16 df-ne 2309 . . . . . . 7  |-  ( B  =/=  D  <->  -.  B  =  D )
1715, 16anbi12i 455 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  D )  <->  ( -.  B  =  C  /\  -.  B  =  D
) )
1814, 17sylnibr 666 . . . . 5  |-  ( ( B  =  C  \/  B  =  D )  ->  -.  ( B  =/= 
C  /\  B  =/=  D ) )
1913, 18anim12i 336 . . . 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 pm4.56 769 . . 3  |-  ( ( -.  ( A  =/= 
C  /\  A  =/=  D )  /\  -.  ( B  =/=  C  /\  B  =/=  D ) )  <->  -.  (
( A  =/=  C  /\  A  =/=  D
)  \/  ( B  =/=  C  /\  B  =/=  D ) ) )
2220, 21syl6ib 160 . 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 2365 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 103    \/ wo 697    = wceq 1331    e. wcel 1480    =/= wne 2308   {cpr 3528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by: (None)
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