Theorem conntop 23360

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Assertion

Ref	Expression
conntop	⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Proof of Theorem conntop

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23356	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3930  ∅c0 4313  {cpr 4608  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  Clsdccld 22959  Conncconn 23354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-conn 23355

This theorem is referenced by:  conncompss  23376  txconn  23632  qtopconn  23652  ufildr  23874  connpconn  35262  cvmliftmolem1  35308  cvmliftmolem2  35309  ordtopconn  36462