Metamath Proof Explorer

Theorem mopn0

Description: The empty set is an open set of a metric space. Part of Theorem T1 of Kreyszig p. 19. (Contributed by NM, 4-Sep-2006)

Ref Expression
Hypothesis mopni.1 
|- J = ( MetOpen ` D )
Assertion mopn0 
|- ( D e. ( *Met ` X ) -> (/) e. J )

Proof

Step Hyp Ref Expression
1 mopni.1 
 |-  J = ( MetOpen ` D )
2 1 mopntop 
 |-  ( D e. ( *Met ` X ) -> J e. Top )
3 0opn 
 |-  ( J e. Top -> (/) e. J )
4 2 3 syl 
 |-  ( D e. ( *Met ` X ) -> (/) e. J )