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Theorem elprn2 45590
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 2944 . . 3 (𝐴𝐶 → ¬ 𝐴 = 𝐶)
21adantl 481 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → ¬ 𝐴 = 𝐶)
3 elpri 4654 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 480 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → (𝐴 = 𝐵𝐴 = 𝐶))
5 orcom 870 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐶𝐴 = 𝐵))
6 df-or 848 . . . 4 ((𝐴 = 𝐶𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
75, 6bitri 275 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶𝐴 = 𝐵))
84, 7sylib 218 . 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 395  wo 847   = wceq 1537  wcel 2106  wne 2938  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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