Theorem ssd 43769

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 1918	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 43764	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966

This theorem is referenced by:  iinssiin  43818  restopnssd  43846  icomnfinre  44265  fnlimfvre  44390  allbutfifvre  44391  limsupresico  44416  liminfresico  44487  limsupgtlem  44493  cnrefiisplem  44545  xlimliminflimsup  44578  rrxsnicc  45016  salrestss  45077  meaiuninclem  45196  meaiininclem  45202  borelmbl  45352  smflimlem1  45487  smflimlem2  45488  smfpimbor1lem1  45514  smfpimbor1lem2  45515  smfsuplem1  45527