Theorem ssd 45660

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 45655	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  45707  restopnssd  45730  icomnfinre  46128  fnlimfvre  46248  allbutfifvre  46249  limsupresico  46274  liminfresico  46345  limsupgtlem  46351  cnrefiisplem  46403  xlimliminflimsup  46436  fourierdlem48  46728  fourierdlem49  46729  rrxsnicc  46874  salrestss  46935  meaiuninclem  47054  meaiininclem  47060  hoicvr  47122  borelmbl  47210  smflimlem1  47345  smflimlem2  47346  smfpimbor1lem1  47372  smfpimbor1lem2  47373  smfsuplem1  47385