Theorem ssd 40011

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypothesis

Ref	Expression
ssd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
ssd	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem ssd

Step	Hyp	Ref	Expression
1		nfv 2010	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 40006	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157   ⊆ wss 3769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-in 3776  df-ss 3783

This theorem is referenced by:  funimassd  40179  icomnfinre  40523  fnlimfvre  40650  allbutfifvre  40651  limsupresico  40676  liminfresico  40747  limsupgtlem  40753  cnrefiisplem  40799  rrxsnicc  41263  meaiuninclem  41440  meaiininclem  41446  borelmbl  41596  smflimlem1  41725  smflimlem2  41726  smfpimbor1lem1  41751  smfpimbor1lem2  41752  smfsuplem1  41763