Theorem ssneldd 3277

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

Hypotheses

Ref	Expression
ssneld.1	⊢ (φ → A ⊆ B)
ssneldd.2	⊢ (φ → ¬ C ∈ B)

Assertion

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

Proof of Theorem ssneldd

Step	Hyp	Ref	Expression
1		ssneldd.2	. 2 ⊢ (φ → ¬ C ∈ B)
2		ssneld.1	. . 3 ⊢ (φ → A ⊆ B)
3	2	ssneld 3276	. 2 ⊢ (φ → (¬ C ∈ B → ¬ C ∈ A))
4	1, 3	mpd 14	1 ⊢ (φ → ¬ C ∈ A)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 1710   ⊆ wss 3258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is used by: (None)