Theorem difxp1ss

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

	Ref	Expression
Assertion	difxp1ss	⊢ ( ( 𝐴 ∖ 𝐶 ) × 𝐵 ) ⊆ ( 𝐴 × 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		difxp1	⊢ ( ( 𝐴 ∖ 𝐶 ) × 𝐵 ) = ( ( 𝐴 × 𝐵 ) ∖ ( 𝐶 × 𝐵 ) )
2		difss	⊢ ( ( 𝐴 × 𝐵 ) ∖ ( 𝐶 × 𝐵 ) ) ⊆ ( 𝐴 × 𝐵 )
3	1 2	eqsstri	⊢ ( ( 𝐴 ∖ 𝐶 ) × 𝐵 ) ⊆ ( 𝐴 × 𝐵 )