Theorem 1stctop 22947

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 2733	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	is1stc 22945	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3	2	simplbi 499	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3948  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  ωcom 7855   ≼ cdom 8937  Topctop 22395  1^stωc1stc 22941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-1stc 22943

This theorem is referenced by:  1stcfb  22949  1stcrest  22957  1stcelcls  22965  lly1stc  23000  1stckgen  23058  tx1stc  23154