Theorem cuspusp 24274

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.

Step	Hyp	Ref	Expression
1		iscusp 24273	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2	1	simplbi 496	1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∅c0 4274  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  TopOpenctopn 17375  Filcfil 23820   fLim cflim 23909  UnifStcuss 24228  UnifSpcusp 24229  CauFil_uccfilu 24260  CUnifSpccusp 24271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-ov 7363  df-cusp 24272

This theorem is referenced by:  cnextucn  24277  ucnextcn  24278  rrhcn  34157  rrhre  34181