Theorem ssinss1d 4143

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 4141	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3896   ∩ cin 4088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-in 4090

This theorem is used by:  exsslsb  34279  ssinss2d  46213  cncfuni  47032  ovolsplit  47134  caragenuncllem  47658  carageniuncllem1  47667  ovnsplit  47794  vonvolmbllem  47806  vonvolmbl  47807