Theorem unss12

Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004)

	Ref	Expression
Assertion	unss12	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( 𝐴 ∪ 𝐶 ) ⊆ ( 𝐵 ∪ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		unss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∪ 𝐶 ) ⊆ ( 𝐵 ∪ 𝐶 ) )
2		unss2	⊢ ( 𝐶 ⊆ 𝐷 → ( 𝐵 ∪ 𝐶 ) ⊆ ( 𝐵 ∪ 𝐷 ) )
3	1 2	sylan9ss	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( 𝐴 ∪ 𝐶 ) ⊆ ( 𝐵 ∪ 𝐷 ) )