Theorem elprn2 44648

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 2944	. . 3 ⊢ (𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶)
2	1	adantl 480	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 = 𝐶)
3		elpri 4649	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4	3	adantr 479	. . 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 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  {cpr 4629

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-v 3474  df-un 3952  df-sn 4628  df-pr 4630

This theorem is referenced by: (None)