This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sslin

Description: Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999)

Ref Expression
Assertion sslin ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∩ 𝐴 ) ⊆ ( 𝐶 ∩ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ssrin ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )
2 incom ⊢ ( 𝐶 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐶 )
3 incom ⊢ ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 )
4 1 2 3 3sstr4g ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∩ 𝐴 ) ⊆ ( 𝐶 ∩ 𝐵 ) )