Theorem nelneq2 2854

Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Assertion

Ref	Expression
nelneq2	⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)

Proof of Theorem nelneq2

Step	Hyp	Ref	Expression
1		eleq2 2818	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2	1	biimpcd 248	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶))
3	2	con3dimp 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720  df-clel 2806

This theorem is referenced by:  nelne1  3036  ssnelpss  4109  opthwiener  5516  ssfin4  10333  pwxpndom2  10688  fzneuz  13614  hauspwpwf1  23890  topdifinffinlem  36826  clsk1indlem1  43475