Theorem conntop 23768

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 2763	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23764	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∩ cin 4092  ∅c0 4281  {cpr 4590  ∪ cuni 4870  ‘cfv 6561  Topctop 23244  Clsdccld 23367  Conncconn 23762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-conn 23763

This theorem is used by:  conncompss  23784  txconn  24041  qtopconn  24061  ufildr  24283  connpconn  36180  cvmliftmolem1  36226  cvmliftmolem2  36227  ordtopconn  37418