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Theorem nelpri 3551
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
nelpri.1  |-  A  =/= 
B
nelpri.2  |-  A  =/= 
C
Assertion
Ref Expression
nelpri  |-  -.  A  e.  { B ,  C }

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2  |-  A  =/= 
B
2 nelpri.2 . 2  |-  A  =/= 
C
3 neanior 2395 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3550 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 621 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 120 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6mp2an 422 1  |-  -.  A  e.  { B ,  C }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480    =/= wne 2308   {cpr 3528
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by:  prneli  3552
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