Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elprn1 Structured version   Visualization version   GIF version

Theorem elprn1 42214
 Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elprn1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)

Proof of Theorem elprn1
StepHypRef Expression
1 neneq 3017 . . 3 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
21adantl 485 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
3 elpri 4561 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 484 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
54ord 861 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (¬ 𝐴 = 𝐵𝐴 = 𝐶))
62, 5mpd 15 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  {cpr 4541 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 2116  ax-9 2124  ax-ext 2794 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 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-v 3471  df-un 3913  df-sn 4540  df-pr 4542 This theorem is referenced by:  fourierdlem70  42757  fourierdlem71  42758  fouriersw  42812  prsal  42899  sge0pr  42972
 Copyright terms: Public domain W3C validator