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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 2825 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ist0 21502 | . 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |
3 | 2 | simplbi 493 | 1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 ∀wral 3117 ∪ cuni 4660 Topctop 21075 Kol2ct0 21488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-uni 4661 df-t0 21495 |
This theorem is referenced by: restt0 21548 sst0 21555 kqt0 21927 t0hmph 21971 kqhmph 22000 ordtopt0 32969 |
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