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| Description: A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.) | 
| Ref | Expression | 
|---|---|
| cuspusp | ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iscusp 24309 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∅c0 4332 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 TopOpenctopn 17467 Filcfil 23854 fLim cflim 23943 UnifStcuss 24263 UnifSpcusp 24264 CauFiluccfilu 24296 CUnifSpccusp 24307 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-cusp 24308 | 
| This theorem is referenced by: cnextucn 24313 ucnextcn 24314 rrhcn 33999 rrhre 34023 | 
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