Theorem xpss2

Description: Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009)

	Ref	Expression
Assertion	xpss2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 × 𝐴 ) ⊆ ( 𝐶 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐶 ⊆ 𝐶
2		xpss12	⊢ ( ( 𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 × 𝐴 ) ⊆ ( 𝐶 × 𝐵 ) )
3	1 2	mpan	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 × 𝐴 ) ⊆ ( 𝐶 × 𝐵 ) )