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Theorem elprn2 41376
 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 2968 . . 3 (𝐴𝐶 → ¬ 𝐴 = 𝐶)
21adantl 474 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → ¬ 𝐴 = 𝐶)
3 elpri 4458 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 473 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
5 orcom 857 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐶𝐴 = 𝐵))
6 df-or 835 . . . 4 ((𝐴 = 𝐶𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
75, 6bitri 267 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
84, 7sylib 210 . 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   ∧ wa 387   ∨ wo 834   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  {cpr 4438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-v 3412  df-un 3829  df-sn 4437  df-pr 4439 This theorem is referenced by: (None)
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