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Theorem elprn2 38701
 Description: A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elprn2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → 𝐴 = 𝐵)

Proof of Theorem elprn2
StepHypRef Expression
1 neneq 2788 . . 3 (𝐴𝐶 → ¬ 𝐴 = 𝐶)
21adantl 481 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → ¬ 𝐴 = 𝐶)
3 elpri 4145 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 480 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
5 orcom 401 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐶𝐴 = 𝐵))
6 df-or 384 . . . 4 ((𝐴 = 𝐶𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
75, 6bitri 263 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
84, 7sylib 207 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (¬ 𝐴 = 𝐶𝐴 = 𝐵))
92, 8mpd 15 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by: (None)
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