Theorem nelpri 3584

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 2414	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 3583	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 622	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 120	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	mp2an 423	1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   = wceq 1335   ∈ wcel 2128   ≠ wne 2327  {cpr 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567

This theorem is referenced by:  prneli  3585  pw1nel3  7166  sucpw1nel3  7168