Theorem 1stctop 22502

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 2738	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	is1stc 22500	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ωcom 7687   ≼ cdom 8689  Topctop 21950  1^stωc1stc 22496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-1stc 22498

This theorem is referenced by:  1stcfb  22504  1stcrest  22512  1stcelcls  22520  lly1stc  22555  1stckgen  22613  tx1stc  22709