Theorem ssd 41716

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 1915	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 41711	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ⊆ wss 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898

This theorem is referenced by:  iinssiin  41764  funimassd  41863  icomnfinre  42189  fnlimfvre  42316  allbutfifvre  42317  limsupresico  42342  liminfresico  42413  limsupgtlem  42419  cnrefiisplem  42471  xlimliminflimsup  42504  rrxsnicc  42942  meaiuninclem  43119  meaiininclem  43125  borelmbl  43275  smflimlem1  43404  smflimlem2  43405  smfpimbor1lem1  43430  smfpimbor1lem2  43431  smfsuplem1  43442