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Theorem 1stctop 21227
 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 2620 . . 3 𝐽 = 𝐽
21is1stc 21225 . 2 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
32simplbi 476 1 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910   ∩ cin 3566  𝒫 cpw 4149  ∪ cuni 4427   class class class wbr 4644  ωcom 7050   ≼ cdom 7938  Topctop 20679  1st𝜔c1stc 21221 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-uni 4428  df-1stc 21223 This theorem is referenced by:  1stcfb  21229  1stcrest  21237  1stcelcls  21245  lly1stc  21280  1stckgen  21338  tx1stc  21434
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