Theorem nelneq2 2866

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 2830	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2	1	biimpcd 249	. 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 1540   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816

This theorem is referenced by:  nelne1  3039  ssnelpss  4114  opthwiener  5519  ssfin4  10350  pwxpndom2  10705  fzneuz  13648  hauspwpwf1  23995  topdifinffinlem  37348  clsk1indlem1  44058