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Theorem nelpr1 4659
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 978 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr1.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4652 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 479 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 256 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 814 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2943 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2936  {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 2698
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 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-v 3473  df-un 3952  df-sn 4631  df-pr 4633
This theorem is referenced by:  cyc3genpmlem  32890  ovnsubadd2lem  46035
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