Theorem neleqtrrd 2328

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
neleqtrrd.1	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
neleqtrrd.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
neleqtrrd	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)

Proof of Theorem neleqtrrd

Step	Hyp	Ref	Expression
1		neleqtrrd.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
2		neleqtrrd.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq2d 2299	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
4	1, 3	mtbird 677	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ∈ wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  tfr1onlemsucaccv  6485  tfrcllemsucaccv  6498  zfz1isolemiso  11056  wrdlndm  11083  structiedg0val  15835