Theorem nelneq2 2904

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 2868	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2	1	biimpcd 241	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶))
3	2	con3dimp 398	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2793  df-clel 2796

This theorem is referenced by:  ssnelpss  3916  opthwiener  5171  ssfin4  9421  pwxpndom2  9776  fzneuz  12674  hauspwpwf1  22118  topdifinffinlem  33692  clsk1indlem1  39120