Theorem conntop 23648

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 2762	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23644	. 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 2145   ∩ cin 3901  ∅c0 4282  {cpr 4589  ∪ cuni 4870  ‘cfv 6537  Topctop 23124  Clsdccld 23247  Conncconn 23642

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-conn 23643

This theorem is used by:  conncompss  23664  txconn  23921  qtopconn  23941  ufildr  24163  connpconn  35822  cvmliftmolem1  35868  cvmliftmolem2  35869  ordtopconn  37066