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Theorem nelprd 4018
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 2750 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 4014 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 141 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 199 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6syl2anc 666 1  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   {cpr 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-v 3084  df-un 3442  df-sn 3998  df-pr 4000
This theorem is referenced by:  renfdisj  9696  sumtp  13803  pmtrprfv3  17088  perfectlem2  24150  perfectALTVlem2  38600  usgra2pthlem1  39009
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