Theorem ssd 45657

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 1934	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 45652	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-ral 3077  df-ss 3921

This theorem is referenced by:  iinssiin  45704  restopnssd  45727  icomnfinre  46125  fnlimfvre  46245  allbutfifvre  46246  limsupresico  46271  liminfresico  46342  limsupgtlem  46348  cnrefiisplem  46400  xlimliminflimsup  46433  fourierdlem48  46725  fourierdlem49  46726  rrxsnicc  46871  salrestss  46932  meaiuninclem  47051  meaiininclem  47057  hoicvr  47119  borelmbl  47207  smflimlem1  47342  smflimlem2  47343  smfpimbor1lem1  47369  smfpimbor1lem2  47370  smfsuplem1  47382