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Theorem t0top 23353
Description: A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
t0top (𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Proof of Theorem t0top
Dummy variables 𝑥 𝑦 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 𝐽 = 𝐽
21ist0 23344 . 2 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
32simplbi 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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