Theorem intss1

Description: An element of a class includes the intersection of the class. Exercise 4 of TakeutiZaring p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995)

	Ref	Expression
Assertion	intss1	$$⊢A\in B\to \bigcap B\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2	1	elint	$$⊢x\in \bigcap B\leftrightarrow \forall y\left(y\in B\to x\in y\right)$$
3		eleq1	$$⊢y=A\to \left(y\in B\leftrightarrow A\in B\right)$$
4		eleq2	$$⊢y=A\to \left(x\in y\leftrightarrow x\in A\right)$$
5	3 4	imbi12d	$$⊢y=A\to \left(\left(y\in B\to x\in y\right)\leftrightarrow \left(A\in B\to x\in A\right)\right)$$
6	5	spcgv	$$⊢A\in B\to \left(\forall y\left(y\in B\to x\in y\right)\to \left(A\in B\to x\in A\right)\right)$$
7	6	pm2.43a	$$⊢A\in B\to \left(\forall y\left(y\in B\to x\in y\right)\to x\in A\right)$$
8	2 7	syl5bi	$$⊢A\in B\to \left(x\in \bigcap B\to x\in A\right)$$
9	8	ssrdv	$$⊢A\in B\to \bigcap B\subseteq A$$