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Theorem nelpri 4655
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 4649 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 154 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 217 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6mp2an 692 1 ¬ 𝐴 ∈ {𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  prneli  4656  ex-dif  30442  ex-in  30444  ex-pss  30447  ex-res  30460  ex-hash  30472
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