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Metamath Proof Explorer

Theorem ssinss1d

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

Ref Expression
Hypothesis ssinss1d.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )
Assertion ssinss1d ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )

Proof

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