Theorem neleqtrd 2861

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 2825	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
4	1, 3	mtbid 324	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814

This theorem is referenced by:  neleqtrrd  2862  smoord  8404  r1tskina  10820  ofccat  15005  mreexexlem2d  17690  opptgdim2  28768  acopyeu  28857  dimlssid  33660  dochnel  41376  stoweidlem26  45982  fourierdlem60  46122  fourierdlem61  46123  sge00  46332  sge0sn  46335  sge0split  46365