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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 2737 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | ist0 23328 | . 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 2108 ∀wral 3061 ∪ cuni 4907 Topctop 22899 Kol2ct0 23314 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-ss 3968 df-uni 4908 df-t0 23321 | 
| This theorem is referenced by: restt0 23374 sst0 23381 kqt0 23754 t0hmph 23798 kqhmph 23827 ordtopt0 36443 | 
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