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Theorem prneli 4180
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
prneli.1 𝐴𝐵
prneli.2 𝐴𝐶
Assertion
Ref Expression
prneli 𝐴 ∉ {𝐵, 𝐶}

Proof of Theorem prneli
StepHypRef Expression
1 prneli.1 . . 3 𝐴𝐵
2 prneli.2 . . 3 𝐴𝐶
31, 2nelpri 4179 . 2 ¬ 𝐴 ∈ {𝐵, 𝐶}
43nelir 2896 1 𝐴 ∉ {𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  wne 2790  wnel 2893  {cpr 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-v 3192  df-un 3565  df-sn 4156  df-pr 4158
This theorem is referenced by:  vdegp1ai  26352
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