|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > regtop | Structured version Visualization version GIF version | ||
| Description: A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) | 
| Ref | Expression | 
|---|---|
| regtop | ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isreg 23341 | . 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ‘cfv 6560 Topctop 22900 clsccl 23027 Regcreg 23318 | 
| 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-ral 3061 df-rex 3070 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-reg 23325 | 
| This theorem is referenced by: regsep2 23385 regr1 23759 kqreg 23760 reghmph 23802 | 
| Copyright terms: Public domain | W3C validator |