Theorem nelprd 4592

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 3029	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4582	. . . 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 591	1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-un 3890  df-sn 4559  df-pr 4561

This theorem is referenced by:  ord2eln012  8426  renfdisj  11200  sumtp  15706  pmtrprfv3  19424  logbgcd1irr  26780  perfectlem2  27215  nbupgrres  29455  usgr2pthlem  29853  eupth2lem3lem6  30325  cycpmco2  33218  cyc2fvx  33219  elrspunsn  33516  esplyind  33771  relogbzexpd  42476  dvrelog2b  42566  dvrelogpow2b  42568  aks4d1p1p4  42571  aks4d1p6  42581  aks6d1c7lem1  42680  mnuprdlem1  44731  mnuprdlem2  44732  perfectALTVlem2  48227