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Theorem nelpri 4340
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 𝐴𝐵
nelpri.2 𝐴𝐶
Assertion
Ref Expression
nelpri ¬ 𝐴 ∈ {𝐵, 𝐶}

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2 𝐴𝐵
2 nelpri.2 . 2 𝐴𝐶
3 neanior 3035 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 4337 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 151 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 207 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6mp2an 672 1 ¬ 𝐴 ∈ {𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  wo 834   = wceq 1631  wcel 2145  wne 2943  {cpr 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-un 3728  df-sn 4317  df-pr 4319
This theorem is referenced by:  prneli  4341  ex-dif  27622  ex-in  27624  ex-pss  27627  ex-res  27640  ex-hash  27652
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