Theorem ssinss1d 42596

Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ssinss1d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Assertion

Ref	Expression
ssinss1d	⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem ssinss1d

Step	Hyp	Ref	Expression
1		ssinss1d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
2		ssinss1 4171	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 3886   ⊆ wss 3887

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-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  ssinss2d  42608  ovolsplit  43529  caragenuncllem  44050  carageniuncllem1  44059  ovnsplit  44186  vonvolmbllem  44198  vonvolmbl  44199