Theorem xpss1

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

	Ref	Expression
Assertion	xpss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 × 𝐶 ) ⊆ ( 𝐵 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐶 ⊆ 𝐶
2		xpss12	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐶 ) → ( 𝐴 × 𝐶 ) ⊆ ( 𝐵 × 𝐶 ) )
3	1 2	mpan2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 × 𝐶 ) ⊆ ( 𝐵 × 𝐶 ) )