Theorem elint

Description: Membership in class intersection. (Contributed by NM, 21-May-1994)

	Ref	Expression
Hypothesis	elint.1	$$⊢A\in \mathrm{V}$$
Assertion	elint	$$⊢A\in \bigcap B\leftrightarrow \forall x\left(x\in B\to A\in x\right)$$

Proof

Step	Hyp	Ref	Expression
1		elint.1	$$⊢A\in \mathrm{V}$$
2		eleq1	$$⊢z=y\to \left(z\in x\leftrightarrow y\in x\right)$$
3	2	imbi2d	$$⊢z=y\to \left(\left(x\in B\to z\in x\right)\leftrightarrow \left(x\in B\to y\in x\right)\right)$$
4	3	albidv	$$⊢z=y\to \left(\forall x\left(x\in B\to z\in x\right)\leftrightarrow \forall x\left(x\in B\to y\in x\right)\right)$$
5		eleq1	$$⊢y=A\to \left(y\in x\leftrightarrow A\in x\right)$$
6	5	imbi2d	$$⊢y=A\to \left(\left(x\in B\to y\in x\right)\leftrightarrow \left(x\in B\to A\in x\right)\right)$$
7	6	albidv	$$⊢y=A\to \left(\forall x\left(x\in B\to y\in x\right)\leftrightarrow \forall x\left(x\in B\to A\in x\right)\right)$$
8		df-int	$$⊢\bigcap B=\left\{z|\forall x\left(x\in B\to z\in x\right)\right\}$$
9	4 7 8	elab2gw	$$⊢A\in \mathrm{V}\to \left(A\in \bigcap B\leftrightarrow \forall x\left(x\in B\to A\in x\right)\right)$$
10	1 9	ax-mp	$$⊢A\in \bigcap B\leftrightarrow \forall x\left(x\in B\to A\in x\right)$$