Theorem nelpri 4619

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 3050	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4611	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 155	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 220	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	mp2an 705	1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {cpr 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590

This theorem is used by:  prneli  4620  ex-dif  30889  ex-in  30891  ex-pss  30894  ex-res  30907  ex-hash  30919