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Theorem nelpr1 4492
Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
nelpr1.a (𝜑𝐴𝑉)
nelpr1.n (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Assertion
Ref Expression
nelpr1 (𝜑𝐴𝐵)

Proof of Theorem nelpr1
StepHypRef Expression
1 nelpr1.n . . 3 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
2 animorrl 973 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr1.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4487 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 481 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 258 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 812 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2989 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842   = wceq 1520  wcel 2079  wne 2982  {cpr 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-v 3434  df-un 3859  df-sn 4467  df-pr 4469
This theorem is referenced by:  cyc3genpmlem  30389  ovnsubadd2lem  42423
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