Theorem toptopon

Description: Alternative definition of Top in terms of TopOn . (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypothesis	toptopon.1	$$⊢X=\bigcup J$$
Assertion	toptopon	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		toptopon.1	$$⊢X=\bigcup J$$
2		istopon	$$⊢J\in \mathrm{TopOn}\left(X\right)\leftrightarrow \left(J\in \mathrm{Top}\wedge X=\bigcup J\right)$$
3	1 2	mpbiran2	$$⊢J\in \mathrm{TopOn}\left(X\right)\leftrightarrow J\in \mathrm{Top}$$
4	3	bicomi	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(X\right)$$