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Theorem nelpri 4620
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 3038 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 4613 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 154 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 216 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6mp2an 691 1 ¬ 𝐴 ∈ {𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944  {cpr 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-un 3920  df-sn 4592  df-pr 4594
This theorem is referenced by:  prneli  4621  ex-dif  29409  ex-in  29411  ex-pss  29414  ex-res  29427  ex-hash  29439
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