Theorem conntop 23341

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 2728	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23337	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3948  ∅c0 4326  {cpr 4634  ∪ cuni 4912  ‘cfv 6553  Topctop 22815  Clsdccld 22940  Conncconn 23335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-conn 23336

This theorem is referenced by:  conncompss  23357  txconn  23613  qtopconn  23633  ufildr  23855  connpconn  34878  cvmliftmolem1  34924  cvmliftmolem2  34925  ordtopconn  35956