Theorem nelpri 4610

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)

Hypotheses

Ref	Expression
nelpri.1	⊢ 𝐴 ≠ 𝐵
nelpri.2	⊢ 𝐴 ≠ 𝐶

Assertion

Ref	Expression
nelpri	⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

Proof of Theorem nelpri

Step	Hyp	Ref	Expression
1		nelpri.1	. 2 ⊢ 𝐴 ≠ 𝐵
2		nelpri.2	. 2 ⊢ 𝐴 ≠ 𝐶
3		neanior 3023	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4602	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 154	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 217	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	mp2an 692	1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  {cpr 4580

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 2115  ax-9 2123  ax-ext 2706

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 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-un 3904  df-sn 4579  df-pr 4581

This theorem is referenced by:  prneli  4611  ex-dif  30447  ex-in  30449  ex-pss  30452  ex-res  30465  ex-hash  30477