Theorem nelpr1 4615

Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
nelpr1.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
nelpr1.n	⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Assertion

Ref	Expression
nelpr1	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem nelpr1

Step	Hyp	Ref	Expression
1		nelpr1.n	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
2		animorrl 996	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		nelpr1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		elprg 4607	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
5	3, 4	syl 18	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	2, 6	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
8	1, 7	mtand 828	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
9	8	neqned 2966	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by:  cyc3genpmlem  33782  ovnsubadd2lem  47860