MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelpr2 Structured version   Visualization version   GIF version

Theorem nelpr2 4582
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 972 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr2.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4578 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 481 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 258 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 812 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
98neqned 3020 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  ovnsubadd2lem  42804
  Copyright terms: Public domain W3C validator