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Theorem nelprd 4038
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
Hypotheses
Ref Expression
nelprd.1  |-  ( ph  ->  A  =/=  B )
nelprd.2  |-  ( ph  ->  A  =/=  C )
Assertion
Ref Expression
nelprd  |-  ( ph  ->  -.  A  e.  { B ,  C }
)

Proof of Theorem nelprd
StepHypRef Expression
1 nelprd.1 . 2  |-  ( ph  ->  A  =/=  B )
2 nelprd.2 . 2  |-  ( ph  ->  A  =/=  C )
3 neanior 2779 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 4036 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 135 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 195 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6syl2anc 659 1  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  renfdisj  9636  pmtrprfv3  16678  perfectlem2  23703  usgra2pthlem1  32725
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