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Theorem nelprd 3981
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 2735 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3976 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 142 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 200 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6syl2anc 673 1  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ 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:  renfdisj  9712  sumtp  13887  pmtrprfv3  17173  perfectlem2  24237  perfectALTVlem2  38989  nbupgrres  39602  eupth2lem3lem6  40145  usgra2pthlem1  40175
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