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Theorem nelpr1 38289
 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 orc 399 . . . . 5 (𝐴 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐶))
21adantl 481 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr1.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4144 . . . . . 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 246 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
8 nelpr1.n . . . 4 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
98adantr 480 . . 3 ((𝜑𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵, 𝐶})
107, 9pm2.65da 598 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
1110neqned 2789 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  ovnsubadd2lem  39535
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