Theorem indisuni

Description: The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Assertion	indisuni	$$⊢\mathrm{I}\left(A\right)=\bigcup \left\{\varnothing ,A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		indislem	$$⊢\left\{\varnothing ,\mathrm{I}\left(A\right)\right\}=\left\{\varnothing ,A\right\}$$
2		fvex	$$⊢\mathrm{I}\left(A\right)\in \mathrm{V}$$
3		indistopon	$$⊢\mathrm{I}\left(A\right)\in \mathrm{V}\to \left\{\varnothing ,\mathrm{I}\left(A\right)\right\}\in \mathrm{TopOn}\left(\mathrm{I}\left(A\right)\right)$$
4	2 3	ax-mp	$$⊢\left\{\varnothing ,\mathrm{I}\left(A\right)\right\}\in \mathrm{TopOn}\left(\mathrm{I}\left(A\right)\right)$$
5	1 4	eqeltrri	$$⊢\left\{\varnothing ,A\right\}\in \mathrm{TopOn}\left(\mathrm{I}\left(A\right)\right)$$
6	5	toponunii	$$⊢\mathrm{I}\left(A\right)=\bigcup \left\{\varnothing ,A\right\}$$