Theorem nelneq 2887

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 2851	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 252	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	con3dimp 413	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-clel 2838

This theorem is used by:  nelne2  3056  onfununi  8324  suc11reg  9584  cantnfp1lem3  9645  oemapvali  9649  mreexmrid  17703  supxrnemnf  33122  elrgspnlem4  33574  elrspunsn  33746  onint1  36988  bj-fvmptunsn2  37930  maxidln0  38724  rencldnfilem  43575  climlimsupcex  46511  icccncfext  46629