Theorem ssd 45020

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 1912	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 45015	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-ral 3060  df-ss 3980

This theorem is referenced by:  iinssiin  45069  restopnssd  45095  icomnfinre  45505  fnlimfvre  45630  allbutfifvre  45631  limsupresico  45656  liminfresico  45727  limsupgtlem  45733  cnrefiisplem  45785  xlimliminflimsup  45818  rrxsnicc  46256  salrestss  46317  meaiuninclem  46436  meaiininclem  46442  borelmbl  46592  smflimlem1  46727  smflimlem2  46728  smfpimbor1lem1  46754  smfpimbor1lem2  46755  smfsuplem1  46767