Theorem ssd 45081

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 1914	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 45076	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-ral 3046  df-ss 3934

This theorem is referenced by:  iinssiin  45130  restopnssd  45153  icomnfinre  45557  fnlimfvre  45679  allbutfifvre  45680  limsupresico  45705  liminfresico  45776  limsupgtlem  45782  cnrefiisplem  45834  xlimliminflimsup  45867  rrxsnicc  46305  salrestss  46366  meaiuninclem  46485  meaiininclem  46491  borelmbl  46641  smflimlem1  46776  smflimlem2  46777  smfpimbor1lem1  46803  smfpimbor1lem2  46804  smfsuplem1  46816