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Theorem nelpri 4546
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 3027 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 4539 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 157 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 220 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6mp2an 692 1 ¬ 𝐴 ∈ {𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 846   = wceq 1542  wcel 2114  wne 2935  {cpr 4519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-v 3401  df-un 3849  df-sn 4518  df-pr 4520
This theorem is referenced by:  prneli  4547  ex-dif  28363  ex-in  28365  ex-pss  28368  ex-res  28381  ex-hash  28393
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