Metamath Proof Explorer


Theorem difxp2ss

Description: Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023)

Ref Expression
Assertion difxp2ss ( 𝐴 × ( 𝐵𝐶 ) ) ⊆ ( 𝐴 × 𝐵 )

Proof

Step Hyp Ref Expression
1 difxp2 ( 𝐴 × ( 𝐵𝐶 ) ) = ( ( 𝐴 × 𝐵 ) ∖ ( 𝐴 × 𝐶 ) )
2 difss ( ( 𝐴 × 𝐵 ) ∖ ( 𝐴 × 𝐶 ) ) ⊆ ( 𝐴 × 𝐵 )
3 1 2 eqsstri ( 𝐴 × ( 𝐵𝐶 ) ) ⊆ ( 𝐴 × 𝐵 )