Theorem 1stctop 23167

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.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	is1stc 23165	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ∩ cin 3946  𝒫 cpw 4601  ∪ cuni 4907   class class class wbr 5147  ωcom 7857   ≼ cdom 8939  Topctop 22615  1^stωc1stc 23161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-pw 4603  df-uni 4908  df-1stc 23163

This theorem is referenced by:  1stcfb  23169  1stcrest  23177  1stcelcls  23185  lly1stc  23220  1stckgen  23278  tx1stc  23374