Theorem ssinss1d 44551

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 4236	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 3943   ⊆ wss 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-in 3951  df-ss 3961

This theorem is referenced by:  ssinss2d  44563  ovolsplit  45511  caragenuncllem  46035  carageniuncllem1  46044  ovnsplit  46171  vonvolmbllem  46183  vonvolmbl  46184