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Theorem 1stctop 22946
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 2732 . . 3 𝐽 = 𝐽
21is1stc 22944 . 2 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
32simplbi 498 1 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3061  wrex 3070  cin 3947  𝒫 cpw 4602   cuni 4908   class class class wbr 5148  ωcom 7854  cdom 8936  Topctop 22394  1stωc1stc 22940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909  df-1stc 22942
This theorem is referenced by:  1stcfb  22948  1stcrest  22956  1stcelcls  22964  lly1stc  22999  1stckgen  23057  tx1stc  23153
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