Theorem nelneq 2861

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 2825	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 249	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	con3dimp 408	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812

This theorem is referenced by:  nelne2  3031  onfununi  8276  suc11reg  9535  cantnfp1lem3  9596  oemapvali  9600  mreexmrid  17604  supxrnemnf  32860  elrgspnlem4  33325  elrspunsn  33508  onint1  36651  bj-fvmptunsn2  37592  maxidln0  38384  rencldnfilem  43270  climlimsupcex  46219  icccncfext  46337