Theorem ssinss2d 45508

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

Hypothesis

Ref	Expression
ssinss2d.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
ssinss2d	⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem ssinss2d

Step	Hyp	Ref	Expression
1		incom 4138	. 2 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		ssinss2d.1	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
3	2	ssinss1d 4175	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ⊆ 𝐶)
4	1, 3	eqsstrid 3953	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 3882   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-in 3890  df-ss 3900

This theorem is referenced by:  caragenuncllem  46955