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Theorem AnelBC 28569
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
AnelBC.1  |-  A  =/= 
B
AnelBC.2  |-  A  =/= 
C
Assertion
Ref Expression
AnelBC  |-  A  e/  { B ,  C }

Proof of Theorem AnelBC
StepHypRef Expression
1 AnelBC.1 . . 3  |-  A  =/= 
B
2 AnelBC.2 . . 3  |-  A  =/= 
C
31, 2nelpri 3837 . 2  |-  -.  A  e.  { B ,  C }
4 df-nel 2604 . 2  |-  ( A  e/  { B ,  C }  <->  -.  A  e.  { B ,  C }
)
53, 4mpbir 202 1  |-  A  e/  { B ,  C }
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1726    =/= wne 2601    e/ wnel 2602   {cpr 3817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823
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