Theorem elprn1 4583

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		elpri 4579	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
2	1	adantr 481	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		neneq 2940	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
4	3	adantl 482	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
5	2, 4	orcnd 884	1 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {cpr 4557

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-un 3888  df-sn 4556  df-pr 4558

This theorem is referenced by:  esplyfvaln  33758  fourierdlem70  46619  fourierdlem71  46620  fouriersw  46674  prsal  46761  sge0pr  46837