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Theorem prneimg 4213
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 4211 . . . . 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 869 . . . . . 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 753 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( A  =  C  \/  A  =  D ) )
4 pm1.4 386 . . . . . . . 8  |-  ( ( B  =  D  \/  B  =  C )  ->  ( B  =  C  \/  B  =  D ) )
54ad2antll 728 . . . . . . 7  |-  ( ( ( ( A  =  C  \/  A  =  D )  /\  ( A  =  C  \/  B  =  C )
)  /\  ( ( B  =  D  \/  A  =  D )  /\  ( B  =  D  \/  B  =  C ) ) )  -> 
( B  =  C  \/  B  =  D ) )
63, 5jca 532 . . . . . 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 488 . . . . . 6  |-  ( -.  ( A  =/=  C  /\  A  =/=  D
)  <->  ( -.  A  =/=  C  \/  -.  A  =/=  D ) )
10 nne 2658 . . . . . . 7  |-  ( -.  A  =/=  C  <->  A  =  C )
11 nne 2658 . . . . . . 7  |-  ( -.  A  =/=  D  <->  A  =  D )
1210, 11orbi12i 521 . . . . . 6  |-  ( ( -.  A  =/=  C  \/  -.  A  =/=  D
)  <->  ( A  =  C  \/  A  =  D ) )
139, 12bitr2i 250 . . . . 5  |-  ( ( A  =  C  \/  A  =  D )  <->  -.  ( A  =/=  C  /\  A  =/=  D
) )
14 ianor 488 . . . . . 6  |-  ( -.  ( B  =/=  C  /\  B  =/=  D
)  <->  ( -.  B  =/=  C  \/  -.  B  =/=  D ) )
15 nne 2658 . . . . . . 7  |-  ( -.  B  =/=  C  <->  B  =  C )
16 nne 2658 . . . . . . 7  |-  ( -.  B  =/=  D  <->  B  =  D )
1715, 16orbi12i 521 . . . . . 6  |-  ( ( -.  B  =/=  C  \/  -.  B  =/=  D
)  <->  ( B  =  C  \/  B  =  D ) )
1814, 17bitr2i 250 . . . . 5  |-  ( ( B  =  C  \/  B  =  D )  <->  -.  ( B  =/=  C  /\  B  =/=  D
) )
1913, 18anbi12i 697 . . . 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 pm4.56 495 . . 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 2670 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 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   {cpr 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035
This theorem is referenced by:  prnebg  4214  symg2bas  16549  m2detleib  19259  usgraexmpldifpr  24526  usgvad2edg  32631  zlmodzxzldeplem  33201
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