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Theorem elprn2 42701
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 2940 . . 3 (𝐴𝐶 → ¬ 𝐴 = 𝐶)
21adantl 485 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → ¬ 𝐴 = 𝐶)
3 elpri 4535 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 484 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
5 orcom 869 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐶𝐴 = 𝐵))
6 df-or 847 . . . 4 ((𝐴 = 𝐶𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
75, 6bitri 278 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
84, 7sylib 221 . 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 399  wo 846   = wceq 1542  wcel 2113  wne 2934  {cpr 4515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-un 3846  df-sn 4514  df-pr 4516
This theorem is referenced by: (None)
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