Theorem nelneq 2858

Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Assertion

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

Proof of Theorem nelneq

Step	Hyp	Ref	Expression
1		eleq1 2822	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 248	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	con3dimp 410	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811

This theorem is referenced by:  nelne2  3041  onfununi  8341  suc11reg  9614  cantnfp1lem3  9675  oemapvali  9679  mreexmrid  17587  supxrnemnf  31981  elrspunsn  32547  onint1  35334  bj-fvmptunsn2  36139  maxidln0  36913  rencldnfilem  41558  climlimsupcex  44485  icccncfext  44603