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

Theorem nelpr2 4552
 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 976 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
3 nelpr2.a . . . . . 6 (𝜑𝐴𝑉)
4 elprg 4546 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4syl 17 . . . . 5 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 484 . . . 4 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
72, 6mpbird 260 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
81, 7mtand 815 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
98neqned 2958 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {cpr 4527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528 This theorem is referenced by:  ovnsubadd2lem  43695
 Copyright terms: Public domain W3C validator