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Theorem elprneb 38857
Description: An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
Assertion
Ref Expression
elprneb  |-  ( ( A  e.  { B ,  C }  /\  B  =/=  C )  ->  ( A  =  B  <->  A  =/=  C ) )

Proof of Theorem elprneb
StepHypRef Expression
1 elpri 3976 . . 3  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
2 neeq1 2705 . . . . . 6  |-  ( B  =  A  ->  ( B  =/=  C  <->  A  =/=  C ) )
32eqcoms 2479 . . . . 5  |-  ( A  =  B  ->  ( B  =/=  C  <->  A  =/=  C ) )
4 pm5.1 875 . . . . . 6  |-  ( ( A  =  B  /\  A  =/=  C )  -> 
( A  =  B  <-> 
A  =/=  C ) )
54ex 441 . . . . 5  |-  ( A  =  B  ->  ( A  =/=  C  ->  ( A  =  B  <->  A  =/=  C ) ) )
63, 5sylbid 223 . . . 4  |-  ( A  =  B  ->  ( B  =/=  C  ->  ( A  =  B  <->  A  =/=  C ) ) )
7 neeq2 2706 . . . . 5  |-  ( A  =  C  ->  ( B  =/=  A  <->  B  =/=  C ) )
8 nesym 2699 . . . . . . . 8  |-  ( B  =/=  A  <->  -.  A  =  B )
9 pm5.1 875 . . . . . . . 8  |-  ( ( A  =  C  /\  -.  A  =  B
)  ->  ( A  =  C  <->  -.  A  =  B ) )
108, 9sylan2b 483 . . . . . . 7  |-  ( ( A  =  C  /\  B  =/=  A )  -> 
( A  =  C  <->  -.  A  =  B
) )
1110necon2abid 2685 . . . . . 6  |-  ( ( A  =  C  /\  B  =/=  A )  -> 
( A  =  B  <-> 
A  =/=  C ) )
1211ex 441 . . . . 5  |-  ( A  =  C  ->  ( B  =/=  A  ->  ( A  =  B  <->  A  =/=  C ) ) )
137, 12sylbird 243 . . . 4  |-  ( A  =  C  ->  ( B  =/=  C  ->  ( A  =  B  <->  A  =/=  C ) ) )
146, 13jaoi 386 . . 3  |-  ( ( A  =  B  \/  A  =  C )  ->  ( B  =/=  C  ->  ( A  =  B  <-> 
A  =/=  C ) ) )
151, 14syl 17 . 2  |-  ( A  e.  { B ,  C }  ->  ( B  =/=  C  ->  ( A  =  B  <->  A  =/=  C ) ) )
1615imp 436 1  |-  ( ( A  e.  { B ,  C }  /\  B  =/=  C )  ->  ( A  =  B  <->  A  =/=  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {cpr 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962
This theorem is referenced by:  dfodd5  38934
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