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Theorem nelprd 3472
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 2342 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 3469 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 597 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 119 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6syl2anc 403 1 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664   = wceq 1289  wcel 1438  wne 2255  {cpr 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453
This theorem is referenced by:  tpfidisj  6636  sumtp  10804
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