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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 2735 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ist0 23344 | . 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |
3 | 2 | simplbi 497 | 1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ∀wral 3059 ∪ cuni 4912 Topctop 22915 Kol2ct0 23330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-ss 3980 df-uni 4913 df-t0 23337 |
This theorem is referenced by: restt0 23390 sst0 23397 kqt0 23770 t0hmph 23814 kqhmph 23843 ordtopt0 36425 |
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