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Theorem nelprd 3523
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 2372 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3520 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 606 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 120 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6syl2anc 408 1  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465    =/= wne 2285   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  tpfidisj  6784  sumtp  11151
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