Theorem elprn1 43174

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} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶)

Proof of Theorem elprn1

Step	Hyp	Ref	Expression
1		neneq 2949	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
2	1	adantl 482	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
3		elpri 4583	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4	3	adantr 481	. . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	ord 861	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (¬ 𝐴 = 𝐵 → 𝐴 = 𝐶))
6	2, 5	mpd 15	1 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {cpr 4563

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by:  fourierdlem70  43717  fourierdlem71  43718  fouriersw  43772  prsal  43859  sge0pr  43932