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Theorem elprn1 45648
Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elprn1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)

Proof of Theorem elprn1
StepHypRef Expression
1 neneq 2946 . . 3 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
21adantl 481 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
3 elpri 4649 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 480 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
54ord 865 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (¬ 𝐴 = 𝐵𝐴 = 𝐶))
62, 5mpd 15 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  fourierdlem70  46191  fourierdlem71  46192  fouriersw  46246  prsal  46333  sge0pr  46409
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