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Theorem AnelBC 27020
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 3565 . 2  |-  -.  A  e.  { B ,  C }
4 df-nel 2415 . 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 5    e. wcel 1621    =/= wne 2412    e/ wnel 2413   {cpr 3545
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-v 2729  df-un 3083  df-sn 3550  df-pr 3551
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