Theorem ssinss1d 38239

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 3803	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 3539   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554

This theorem is referenced by:  ssinss2d  38253  ovolsplit  38881  caragenuncllem  39402  carageniuncllem1  39411  ovnsplit  39538  vonvolmbllem  39550  vonvolmbl  39551