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Metamath Proof Explorer

Theorem ssrind

Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022)

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

Proof

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