Theorem dfss5

Description: Alternate definition of subclass relationship: a class A is a subclass of another class B iff each element of A is equal to an element of B . (Contributed by AV, 13-Nov-2020)

	Ref	Expression
Assertion	dfss5	|- ( A C_ B <-> A. x e. A E. y e. B x = y )

Proof

Step	Hyp	Ref	Expression
1		dfss3	|- ( A C_ B <-> A. x e. A x e. B )
2		clel5	|- ( x e. B <-> E. y e. B x = y )
3	2	ralbii	|- ( A. x e. A x e. B <-> A. x e. A E. y e. B x = y )
4	1 3	bitri	|- ( A C_ B <-> A. x e. A E. y e. B x = y )