Theorem prtlem80 40109

Description: Lemma for prter2 40129. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Assertion

Ref	Expression
prtlem80	⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))

Proof of Theorem prtlem80

Step	Hyp	Ref	Expression
1		neldifsnd 4756	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ∖ cdif 4048  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-sn 4589

This theorem is used by: (None)