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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 3024 . . 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 216 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6syl2anc 582 1 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2929  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-un 3949  df-sn 4631  df-pr 4633
This theorem is referenced by:  ord2eln012  8518  renfdisj  11306  sumtp  15731  pmtrprfv3  19421  logbgcd1irr  26771  perfectlem2  27208  nbupgrres  29249  usgr2pthlem  29649  eupth2lem3lem6  30115  cycpmco2  32946  cyc2fvx  32947  elrspunsn  33241  relogbzexpd  41574  dvrelog2b  41666  dvrelogpow2b  41668  aks4d1p1p4  41671  aks4d1p6  41681  aks6d1c7lem1  41780  mnuprdlem1  43848  mnuprdlem2  43849  perfectALTVlem2  47196
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