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Theorem elprn2 43146
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 2951 . . 3 (𝐴𝐶 → ¬ 𝐴 = 𝐶)
21adantl 482 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → ¬ 𝐴 = 𝐶)
3 elpri 4589 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 481 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
5 orcom 867 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐶𝐴 = 𝐵))
6 df-or 845 . . . 4 ((𝐴 = 𝐶𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
75, 6bitri 274 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
84, 7sylib 217 . 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 396  wo 844   = wceq 1542  wcel 2110  wne 2945  {cpr 4569
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 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-v 3433  df-un 3897  df-sn 4568  df-pr 4570
This theorem is referenced by: (None)
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