Theorem nelpr2 4603

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 980	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
3		nelpr2.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		elprg 4596	. . . . . 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 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	2, 6	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
8	1, 7	mtand 815	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
9	8	neqned 2935	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  {cpr 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-un 3902  df-sn 4574  df-pr 4576

This theorem is referenced by:  chnccat  18532  ovnsubadd2lem  46691