Theorem nelpr2 4585

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

Hypotheses

Ref	Expression
nelpr2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
nelpr2.n	⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Assertion

Ref	Expression
nelpr2	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem nelpr2

Step	Hyp	Ref	Expression
1		nelpr2.n	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
2		animorr 986	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		nelpr2.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		elprg 4578	. . . . . 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 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	2, 6	mpbird 258	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
8	1, 7	mtand 821	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
9	8	neqned 2941	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {cpr 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-un 3888  df-sn 4556  df-pr 4558

This theorem is referenced by:  chnccat  18583  ovnsubadd2lem  47088