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Theorem t0top 23277
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 2725 . . 3 𝐽 = 𝐽
21ist0 23268 . 2 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
32simplbi 496 1 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wral 3050   cuni 4909  Topctop 22839  Kol2ct0 23254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-ss 3961  df-uni 4910  df-t0 23261
This theorem is referenced by:  restt0  23314  sst0  23321  kqt0  23694  t0hmph  23738  kqhmph  23767  ordtopt0  36057
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