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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 21937 | . 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ‘cfv 6324 Topctop 21498 clsccl 21623 Regcreg 21914 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-reg 21921 |
This theorem is referenced by: regsep2 21981 regr1 22355 kqreg 22356 reghmph 22398 |
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