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Theorem nelpr1 37499
Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
nelpr1.a  |-  ( ph  ->  A  e.  V )
nelpr1.n  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Assertion
Ref Expression
nelpr1  |-  ( ph  ->  A  =/=  B )

Proof of Theorem nelpr1
StepHypRef Expression
1 orc 392 . . . . 5  |-  ( A  =  B  ->  ( A  =  B  \/  A  =  C )
)
21adantl 473 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  =  B  \/  A  =  C )
)
3 nelpr1.a . . . . . 6  |-  ( ph  ->  A  e.  V )
4 elprg 3975 . . . . . 6  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
53, 4syl 17 . . . . 5  |-  ( ph  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
) )
65adantr 472 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
72, 6mpbird 240 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  e.  { B ,  C } )
8 nelpr1.n . . . 4  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
98adantr 472 . . 3  |-  ( (
ph  /\  A  =  B )  ->  -.  A  e.  { B ,  C } )
107, 9pm2.65da 586 . 2  |-  ( ph  ->  -.  A  =  B )
1110neqned 2650 1  |-  ( ph  ->  A  =/=  B )
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:  ovnsubadd2lem  38585
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