Theorem eltop

Description: Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006)

	Ref	Expression
Assertion	eltop	$$⊢J\in \mathrm{Top}\to \left(A\in J\leftrightarrow A\subseteq \bigcup \left(J\cap 𝒫A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tgtop	$$⊢J\in \mathrm{Top}\to \mathrm{topGen}\left(J\right)=J$$
2	1	eleq2d	$$⊢J\in \mathrm{Top}\to \left(A\in \mathrm{topGen}\left(J\right)\leftrightarrow A\in J\right)$$
3		eltg	$$⊢J\in \mathrm{Top}\to \left(A\in \mathrm{topGen}\left(J\right)\leftrightarrow A\subseteq \bigcup \left(J\cap 𝒫A\right)\right)$$
4	2 3	bitr3d	$$⊢J\in \mathrm{Top}\to \left(A\in J\leftrightarrow A\subseteq \bigcup \left(J\cap 𝒫A\right)\right)$$