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Theorem cuspusp 22085
Description: A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
cuspusp (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Proof of Theorem cuspusp
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 iscusp 22084 . 2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
21simplbi 476 1 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wne 2791  wral 2909  c0 3907  cfv 5876  (class class class)co 6635  Basecbs 15838  TopOpenctopn 16063  Filcfil 21630   fLim cflim 21719  UnifStcuss 22038  UnifSpcusp 22039  CauFiluccfilu 22071  CUnifSpccusp 22082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-cusp 22083
This theorem is referenced by:  cnextucn  22088  ucnextcn  22089  rrhcn  30015  rrhre  30039
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