Theorem dfss7

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

	Ref	Expression
Assertion	dfss7	$$⊢B\subseteq A\leftrightarrow \left\{x\in A|x\in B\right\}=B$$

Proof

Step	Hyp	Ref	Expression
1		dfss2	$$⊢B\subseteq A\leftrightarrow B\cap A=B$$
2		dfin5	$$⊢A\cap B=\left\{x\in A|x\in B\right\}$$
3	2	ineqcomi	$$⊢B\cap A=\left\{x\in A|x\in B\right\}$$
4	3	eqeq1i	$$⊢B\cap A=B\leftrightarrow \left\{x\in A|x\in B\right\}=B$$
5	1 4	bitri	$$⊢B\subseteq A\leftrightarrow \left\{x\in A|x\in B\right\}=B$$