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Theorem nelpr2 40717
 Description: If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
nelpr2.a (𝜑𝐴𝑉)
nelpr2.n (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Assertion
Ref Expression
nelpr2 (𝜑𝐴𝐶)

Proof of Theorem nelpr2
StepHypRef Expression
1 nelpr2.n . . 3 (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
2 animorr 961 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr2.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4456 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 473 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 249 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 803 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
98neqned 2968 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   = wceq 1507   ∈ wcel 2048   ≠ wne 2961  {cpr 4437 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-v 3411  df-un 3830  df-sn 4436  df-pr 4438 This theorem is referenced by:  ovnsubadd2lem  42304
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