Theorem elprn2 43065

Description: A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elprn2	⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → 𝐴 = 𝐵)

Proof of Theorem elprn2

Step	Hyp	Ref	Expression
1		neneq 2948	. . 3 ⊢ (𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶)
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 = 𝐶)
3		elpri 4580	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5		orcom 866	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐵))
6		df-or 844	. . . 4 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵))
7	5, 6	bitri 274	. . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵))
8	4, 7	sylib 217	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵))
9	2, 8	mpd 15	1 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-un 3888  df-sn 4559  df-pr 4561

This theorem is referenced by: (None)