Theorem nelpr1 4492

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 973	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		nelpr1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		elprg 4487	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	2, 6	mpbird 258	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶})
8	1, 7	mtand 812	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
9	8	neqned 2989	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1520   ∈ wcel 2079   ≠ wne 2982  {cpr 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-v 3434  df-un 3859  df-sn 4467  df-pr 4469

This theorem is referenced by:  cyc3genpmlem  30389  ovnsubadd2lem  42423