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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 21943 | . 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ⊆ wss 3939 ‘cfv 6358 Topctop 21504 clsccl 21629 Regcreg 21920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-reg 21927 |
This theorem is referenced by: regsep2 21987 regr1 22361 kqreg 22362 reghmph 22404 |
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