Theorem nelpri 4340

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 3035	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4337	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 151	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 207	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	mp2an 672	1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

Syntax hints:  ¬ wn 3   ∧ wa 382   ∨ wo 834   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  {cpr 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-un 3728  df-sn 4317  df-pr 4319

This theorem is referenced by:  prneli  4341  ex-dif  27622  ex-in  27624  ex-pss  27627  ex-res  27640  ex-hash  27652