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Theorem nelpr1 39576
 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 507 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr1.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4229 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 480 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 247 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 692 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2830 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {cpr 4212 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213 This theorem is referenced by:  ovnsubadd2lem  41180
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