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 3037	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 4583	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 154	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 216	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	syl2anc 584	1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by:  ord2eln012  8327  renfdisj  11035  sumtp  15461  pmtrprfv3  19062  logbgcd1irr  25944  perfectlem2  26378  nbupgrres  27731  usgr2pthlem  28131  eupth2lem3lem6  28597  cycpmco2  31400  cyc2fvx  31401  relogbzexpd  39983  dvrelog2b  40074  dvrelogpow2b  40076  aks4d1p1p4  40079  aks4d1p6  40089  mnuprdlem1  41890  mnuprdlem2  41891  perfectALTVlem2  45174