Theorem nelprd 3472

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Hypotheses

Ref	Expression
nelprd.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
nelprd.2	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Assertion

Ref	Expression
nelprd	⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Proof of Theorem nelprd

Step	Hyp	Ref	Expression
1		nelprd.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		nelprd.2	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
3		neanior 2342	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 3469	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 597	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 119	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	syl2anc 403	1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 664   = wceq 1289   ∈ wcel 1438   ≠ wne 2255  {cpr 3447

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453

This theorem is referenced by:  tpfidisj  6636  sumtp  10804