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Theorem nelpri 3835
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 2689 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3834 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 129 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 188 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6mp2an 654 1  |-  -.  A  e.  { B ,  C }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {cpr 3815
This theorem is referenced by:  constr3pthlem1  21642  konigsberg  21709  ex-dif  21731  ex-in  21733  ex-pss  21736  ex-res  21749  AnelBC  28507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821
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