Theorem flimelbas

Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009) (Revised by Mario Carneiro, 9-Apr-2015)

	Ref	Expression
Hypothesis	flimuni.1	$$⊢X=\bigcup J$$
Assertion	flimelbas	$$⊢A\in \left(J\mathrm{fLim}F\right)\to A\in X$$

Proof

Step	Hyp	Ref	Expression
1		flimuni.1	$$⊢X=\bigcup J$$
2	1	elflim2	$$⊢A\in \left(J\mathrm{fLim}F\right)\leftrightarrow \left(\left(J\in \mathrm{Top}\wedge F\in \bigcup \mathrm{ran}\mathrm{Fil}\wedge F\subseteq 𝒫X\right)\wedge \left(A\in X\wedge \mathrm{nei}\left(J\right)\left(\left\{A\right\}\right)\subseteq F\right)\right)$$
3	2	simprbi	$$⊢A\in \left(J\mathrm{fLim}F\right)\to \left(A\in X\wedge \mathrm{nei}\left(J\right)\left(\left\{A\right\}\right)\subseteq F\right)$$
4	3	simpld	$$⊢A\in \left(J\mathrm{fLim}F\right)\to A\in X$$