Theorem 1stctop 23358

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 2731	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	is1stc 23356	. 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 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3896  𝒫 cpw 4547  ∪ cuni 4856   class class class wbr 5089  ωcom 7796   ≼ cdom 8867  Topctop 22808  1^stωc1stc 23352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-ss 3914  df-pw 4549  df-uni 4857  df-1stc 23354

This theorem is referenced by:  1stcfb  23360  1stcrest  23368  1stcelcls  23376  lly1stc  23411  1stckgen  23469  tx1stc  23565