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} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶)

Proof of Theorem elprn1

Step	Hyp	Ref	Expression
1		neneq 2946	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
3		elpri 4649	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	ord 865	. 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 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