Theorem nelneq 2865

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 2829	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 251	. 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 1548   ∈ wcel 2121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-clel 2816

This theorem is referenced by:  nelne2  3034  onfununi  8275  suc11reg  9535  cantnfp1lem3  9596  oemapvali  9600  mreexmrid  17604  supxrnemnf  32864  elrgspnlem4  33330  elrspunsn  33516  onint1  36692  bj-fvmptunsn2  37633  maxidln0  38427  rencldnfilem  43280  climlimsupcex  46226  icccncfext  46344