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Theorem nelprd 3618
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 2434 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3615 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 632 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 121 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6syl2anc 411 1  |-  ( ph  ->  -.  A  e.  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148    =/= wne 2347   {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  tpfidisj  6926  sumtp  11417
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