Theorem 1stctop 23503

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 23501	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3	2	simplbi 500	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∩ cin 3903  𝒫 cpw 4555  ∪ cuni 4865   class class class wbr 5100  ωcom 7846   ≼ cdom 8925  Topctop 22953  1^stωc1stc 23497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-ss 3921  df-pw 4557  df-uni 4866  df-1stc 23499

This theorem is referenced by:  1stcfb  23505  1stcrest  23513  1stcelcls  23521  lly1stc  23556  1stckgen  23614  tx1stc  23710