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	$$⊢A\subseteq B\leftrightarrow \forall x\in A\exists y\in Bx=y$$

Proof

Step	Hyp	Ref	Expression
1		dfss3	$$⊢A\subseteq B\leftrightarrow \forall x\in Ax\in B$$
2		clel5	$$⊢x\in B\leftrightarrow \exists y\in Bx=y$$
3	2	ralbii	$$⊢\forall x\in Ax\in B\leftrightarrow \forall x\in A\exists y\in Bx=y$$
4	1 3	bitri	$$⊢A\subseteq B\leftrightarrow \forall x\in A\exists y\in Bx=y$$