Theorem difxp2ss

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

	Ref	Expression
Assertion	difxp2ss	⊢ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ⊆ ( 𝐴 × 𝐵 )

Proof

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