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Theorem prneimg 4152
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 4150 . . . . 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 870 . . . . . 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 752 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( A  =  C  \/  A  =  D ) )
4 pm1.4 384 . . . . . . . 8  |-  ( ( B  =  D  \/  B  =  C )  ->  ( B  =  C  \/  B  =  D ) )
54ad2antll 727 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( B  =  C  \/  B  =  D ) )
63, 5jca 530 . . . . . 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 195 . . . . 5  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  -> 
( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
) )
81, 7syl6bi 228 . . . 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 ianor 486 . . . . . 6  |-  ( -.  ( A  =/=  C  /\  A  =/=  D
)  <->  ( -.  A  =/=  C  \/  -.  A  =/=  D ) )
10 nne 2604 . . . . . . 7  |-  ( -.  A  =/=  C  <->  A  =  C )
11 nne 2604 . . . . . . 7  |-  ( -.  A  =/=  D  <->  A  =  D )
1210, 11orbi12i 519 . . . . . 6  |-  ( ( -.  A  =/=  C  \/  -.  A  =/=  D
)  <->  ( A  =  C  \/  A  =  D ) )
139, 12bitr2i 250 . . . . 5  |-  ( ( A  =  C  \/  A  =  D )  <->  -.  ( A  =/=  C  /\  A  =/=  D
) )
14 ianor 486 . . . . . 6  |-  ( -.  ( B  =/=  C  /\  B  =/=  D
)  <->  ( -.  B  =/=  C  \/  -.  B  =/=  D ) )
15 nne 2604 . . . . . . 7  |-  ( -.  B  =/=  C  <->  B  =  C )
16 nne 2604 . . . . . . 7  |-  ( -.  B  =/=  D  <->  B  =  D )
1715, 16orbi12i 519 . . . . . 6  |-  ( ( -.  B  =/=  C  \/  -.  B  =/=  D
)  <->  ( B  =  C  \/  B  =  D ) )
1814, 17bitr2i 250 . . . . 5  |-  ( ( B  =  C  \/  B  =  D )  <->  -.  ( B  =/=  C  /\  B  =/=  D
) )
1913, 18anbi12i 695 . . . 4  |-  ( ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D ) )  <->  ( -.  ( A  =/=  C  /\  A  =/=  D
)  /\  -.  ( B  =/=  C  /\  B  =/=  D ) ) )
208, 19syl6ib 226 . . 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 pm4.56 493 . . 3  |-  ( ( -.  ( A  =/= 
C  /\  A  =/=  D )  /\  -.  ( B  =/=  C  /\  B  =/=  D ) )  <->  -.  (
( A  =/=  C  /\  A  =/=  D
)  \/  ( B  =/=  C  /\  B  =/=  D ) ) )
2220, 21syl6ib 226 . 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 2616 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {cpr 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3060  df-un 3418  df-sn 3972  df-pr 3974
This theorem is referenced by:  prnebg  4153  symg2bas  16745  m2detleib  19423  usgraexmpldifpr  24804  usgvad2edg  38021  zlmodzxzldeplem  38591
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