Theorem neleqtrd 2889

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
neleqtrd.1	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
neleqtrd.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
neleqtrd	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Proof of Theorem neleqtrd

Step	Hyp	Ref	Expression
1		neleqtrd.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
2		neleqtrd.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq2d 2853	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
4	1, 3	mtbid 316	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1508   ∈ wcel 2051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2752

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2773  df-clel 2848

This theorem is referenced by:  neleqtrrd  2890  smoord  7812  r1tskina  10008  ofccat  14196  mreexexlem2d  16786  opptgdim2  26248  acopyeu  26338  dochnel  38014  stoweidlem26  41777  fourierdlem60  41917  fourierdlem61  41918  sge00  42124  sge0sn  42127  sge0split  42157  iundjiunlem  42207