Theorem nelpr1 4630

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 982	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		nelpr1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		elprg 4624	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	2, 6	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
8	1, 7	mtand 815	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
9	8	neqned 2939	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  {cpr 4603

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-un 3931  df-sn 4602  df-pr 4604

This theorem is referenced by:  cyc3genpmlem  33162  ovnsubadd2lem  46674