Theorem unss2

Description: Subclass law for union of classes. Exercise 7 of TakeutiZaring p. 18. (Contributed by NM, 14-Oct-1999)

	Ref	Expression
Assertion	unss2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∪ 𝐴 ) ⊆ ( 𝐶 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		unss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∪ 𝐶 ) ⊆ ( 𝐵 ∪ 𝐶 ) )
2		uncom	⊢ ( 𝐶 ∪ 𝐴 ) = ( 𝐴 ∪ 𝐶 )
3		uncom	⊢ ( 𝐶 ∪ 𝐵 ) = ( 𝐵 ∪ 𝐶 )
4	1 2 3	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∪ 𝐴 ) ⊆ ( 𝐶 ∪ 𝐵 ) )