Theorem dfss7

Description: Alternate definition of subclass relationship. (Contributed by AV, 1-Aug-2022)

	Ref	Expression
Assertion	dfss7	|- ( B C_ A <-> { x e. A | x e. B } = B )

Proof

Step	Hyp	Ref	Expression
1		df-ss	|- ( B C_ A <-> ( B i^i A ) = B )
2		incom	|- ( B i^i A ) = ( A i^i B )
3		dfin5	|- ( A i^i B ) = { x e. A | x e. B }
4	2 3	eqtri	|- ( B i^i A ) = { x e. A | x e. B }
5	4	eqeq1i	|- ( ( B i^i A ) = B <-> { x e. A | x e. B } = B )
6	1 5	bitri	|- ( B C_ A <-> { x e. A | x e. B } = B )