Metamath Proof Explorer

Theorem cuspusp

Description: A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017)

Ref Expression
Assertion cuspusp 
|- ( W e. CUnifSp -> W e. UnifSp )

Proof

Step Hyp Ref Expression
1 iscusp 
 |-  ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. ( CauFilU ` ( UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
2 1 simplbi 
 |-  ( W e. CUnifSp -> W e. UnifSp )