Theorem elinlem

Description: Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020)

	Ref	Expression
Assertion	elinlem	$$⊢A\in \left(B\cap C\right)\leftrightarrow \left(A\in B\wedge \mathrm{I}\left(A\right)\in C\right)$$

Proof

Step	Hyp	Ref	Expression
1		elin	$$⊢A\in \left(B\cap C\right)\leftrightarrow \left(A\in B\wedge A\in C\right)$$
2		fvi	$$⊢A\in B\to \mathrm{I}\left(A\right)=A$$
3	2	eqcomd	$$⊢A\in B\to A=\mathrm{I}\left(A\right)$$
4	3	eleq1d	$$⊢A\in B\to \left(A\in C\leftrightarrow \mathrm{I}\left(A\right)\in C\right)$$
5	4	pm5.32i	$$⊢\left(A\in B\wedge A\in C\right)\leftrightarrow \left(A\in B\wedge \mathrm{I}\left(A\right)\in C\right)$$
6	1 5	bitri	$$⊢A\in \left(B\cap C\right)\leftrightarrow \left(A\in B\wedge \mathrm{I}\left(A\right)\in C\right)$$