Theorem ssd 42611

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 1917	. 2 ⊢ Ⅎ𝑥𝜑
2		ssd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	1, 2	ssdf 42606	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ⊆ wss 3886

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3431  df-in 3893  df-ss 3903

This theorem is referenced by:  iinssiin  42659  funimassd  42751  icomnfinre  43071  fnlimfvre  43196  allbutfifvre  43197  limsupresico  43222  liminfresico  43293  limsupgtlem  43299  cnrefiisplem  43351  xlimliminflimsup  43384  rrxsnicc  43822  meaiuninclem  43999  meaiininclem  44005  borelmbl  44155  smflimlem1  44284  smflimlem2  44285  smfpimbor1lem1  44310  smfpimbor1lem2  44311  smfsuplem1  44322