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Theorem nelprd 4661
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 3032 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 4653 . . . 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, 6syl2anc 584 1 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-un 3967  df-sn 4631  df-pr 4633
This theorem is referenced by:  ord2eln012  8533  renfdisj  11318  sumtp  15781  pmtrprfv3  19486  logbgcd1irr  26851  perfectlem2  27288  nbupgrres  29395  usgr2pthlem  29795  eupth2lem3lem6  30261  cycpmco2  33135  cyc2fvx  33136  elrspunsn  33436  relogbzexpd  41956  dvrelog2b  42047  dvrelogpow2b  42049  aks4d1p1p4  42052  aks4d1p6  42062  aks6d1c7lem1  42161  mnuprdlem1  44267  mnuprdlem2  44268  perfectALTVlem2  47646
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