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Theorem nelpri 3661
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 2531 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3660 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 127 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 187 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6mp2an 653 1  |-  -.  A  e.  { B ,  C }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {cpr 3641
This theorem is referenced by:  ex-dif  20810  ex-in  20812  ex-pss  20815  ex-res  20828  konigsberg  23911  ftp  26893  AnelBC  28234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647
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