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Theorem nelpr2 4648
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 975 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr2.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4642 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 480 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 257 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 813 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
98neqned 2939 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wcel 2098  wne 2932  {cpr 4623
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 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-un 3946  df-sn 4622  df-pr 4624
This theorem is referenced by:  ovnsubadd2lem  45907
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