Theorem ssequn2

Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994)

	Ref	Expression
Assertion	ssequn2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∪ 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssequn1	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 )
2		uncom	⊢ ( 𝐴 ∪ 𝐵 ) = ( 𝐵 ∪ 𝐴 )
3	2	eqeq1i	⊢ ( ( 𝐴 ∪ 𝐵 ) = 𝐵 ↔ ( 𝐵 ∪ 𝐴 ) = 𝐵 )
4	1 3	bitri	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∪ 𝐴 ) = 𝐵 )