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Theorem nelprd 4424
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
Hypotheses
Ref Expression
nelprd.1 (𝜑𝐴𝐵)
nelprd.2 (𝜑𝐴𝐶)
Assertion
Ref Expression
nelprd (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Proof of Theorem nelprd
StepHypRef Expression
1 nelprd.1 . 2 (𝜑𝐴𝐵)
2 nelprd.2 . 2 (𝜑𝐴𝐶)
3 neanior 3091 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 4419 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 152 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 209 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6syl2anc 581 1 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 880   = wceq 1658  wcel 2166  wne 2999  {cpr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-un 3803  df-sn 4398  df-pr 4400
This theorem is referenced by:  renfdisj  10417  sumtp  14855  pmtrprfv3  18224  logbgcd1irr  24934  perfectlem2  25368  nbupgrres  26661  usgr2pthlem  27065  eupth2lem3lem6  27610  perfectALTVlem2  42461
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