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Theorem nelpr1 4551
 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 979 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr1.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4544 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 485 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 260 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 816 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2959 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  {cpr 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-v 3412  df-un 3864  df-sn 4524  df-pr 4526 This theorem is referenced by:  cyc3genpmlem  30945  ovnsubadd2lem  43651
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