Theorem nelprd 4618

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 3052	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4608	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 154	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 219	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	syl2anc 593	1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  {cpr 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-v 3458  df-un 3911  df-sn 4585  df-pr 4587

This theorem is referenced by:  ord2eln012  8468  renfdisj  11244  sumtp  15778  pmtrprfv3  19496  logbgcd1irr  26861  perfectlem2  27296  nbupgrres  29567  usgr2pthlem  29965  eupth2lem3lem6  30437  cycpmco2  33315  cyc2fvx  33316  elrspunsn  33617  esplyind  33874  relogbzexpd  42598  dvrelog2b  42688  dvrelogpow2b  42690  aks4d1p1p4  42693  aks4d1p6  42703  aks6d1c7lem1  42802  mnuprdlem1  44853  mnuprdlem2  44854  perfectALTVlem2  48349