Theorem elprn2 38701

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 2788	. . 3 ⊢ (𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶)
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 = 𝐶)
3		elpri 4145	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5		orcom 401	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐵))
6		df-or 384	. . . 4 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵))
7	5, 6	bitri 263	. . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵))
8	4, 7	sylib 207	. 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   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128

This theorem is referenced by: (None)