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Theorem elprn1 43881
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 2950 . . 3 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
21adantl 483 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
3 elpri 4609 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 482 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
54ord 863 . 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 397  wo 846   = wceq 1542  wcel 2107  wne 2944  {cpr 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3448  df-un 3916  df-sn 4588  df-pr 4590
This theorem is referenced by:  fourierdlem70  44424  fourierdlem71  44425  fouriersw  44479  prsal  44566  sge0pr  44642
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