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Theorem 1stctop 22345
Description: A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Assertion
Ref Expression
1stctop (𝐽 ∈ 1stω → 𝐽 ∈ Top)

Proof of Theorem 1stctop
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 𝐽 = 𝐽
21is1stc 22343 . 2 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
32simplbi 501 1 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2110  wral 3061  wrex 3062  cin 3870  𝒫 cpw 4518   cuni 4824   class class class wbr 5058  ωcom 7649  cdom 8629  Topctop 21795  1stωc1stc 22339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-in 3878  df-ss 3888  df-pw 4520  df-uni 4825  df-1stc 22341
This theorem is referenced by:  1stcfb  22347  1stcrest  22355  1stcelcls  22363  lly1stc  22398  1stckgen  22456  tx1stc  22552
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