Theorem 1stctop 23611

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 2762	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	is1stc 23609	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ∩ cin 3903  𝒫 cpw 4561  ∪ cuni 4871   class class class wbr 5108  ωcom 7860   ≼ cdom 8939  Topctop 23061  1^stωc1stc 23605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-ss 3921  df-pw 4563  df-uni 4872  df-1stc 23607

This theorem is used by:  1stcfb  23613  1stcrest  23621  1stcelcls  23629  lly1stc  23664  1stckgen  23722  tx1stc  23818