Theorem ssneld 3276

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

Hypothesis

Ref	Expression
ssneld.1	⊢ (φ → A ⊆ B)

Assertion

Ref	Expression
ssneld	⊢ (φ → (¬ C ∈ B → ¬ C ∈ A))

Proof of Theorem ssneld

Step	Hyp	Ref	Expression
1		ssneld.1	. . 3 ⊢ (φ → A ⊆ B)
2	1	sseld 3273	. 2 ⊢ (φ → (C ∈ A → C ∈ B))
3	2	con3d 125	1 ⊢ (φ → (¬ C ∈ B → ¬ C ∈ A))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1710   ⊆ wss 3258

This theorem was proved from 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 theorem 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 referenced by:  ssneldd  3277