Theorem neleqtrd 2859

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 2823	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
4	1, 3	mtbid 324	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = 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:  neleqtrrd  2860  smoord  8307  r1tskina  10705  ofccat  14904  mreexexlem2d  17580  chnccat  18561  opptgdim2  28829  acopyeu  28918  dimlssid  33810  dochnel  41769  stoweidlem26  46384  fourierdlem60  46524  fourierdlem61  46525  sge00  46734  sge0sn  46737  sge0split  46767