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Mirrors > Home > MPE Home > Th. List > t0top | Structured version Visualization version GIF version |
Description: A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
t0top | ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ist0 21925 | . 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |
3 | 2 | simplbi 501 | 1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 ∀wral 3106 ∪ cuni 4800 Topctop 21498 Kol2ct0 21911 |
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-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-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-t0 21918 |
This theorem is referenced by: restt0 21971 sst0 21978 kqt0 22351 t0hmph 22395 kqhmph 22424 ordtopt0 33903 |
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