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		df-ss	$$⊢B\subseteq A\leftrightarrow B\cap A=B$$
2		incom	$$⊢B\cap A=A\cap B$$
3		dfin5	$$⊢A\cap B=\left\{x\in A|x\in B\right\}$$
4	2 3	eqtri	$$⊢B\cap A=\left\{x\in A|x\in B\right\}$$
5	4	eqeq1i	$$⊢B\cap A=B\leftrightarrow \left\{x\in A|x\in B\right\}=B$$
6	1 5	bitri	$$⊢B\subseteq A\leftrightarrow \left\{x\in A|x\in B\right\}=B$$