Theorem conntop 23611

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 2766	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23607	. 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 2146   ∩ cin 3907  ∅c0 4289  {cpr 4596  ∪ cuni 4877  ‘cfv 6543  Topctop 23087  Clsdccld 23210  Conncconn 23605

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 2148  ax-9 2156  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-conn 23606

This theorem is used by:  conncompss  23627  txconn  23883  qtopconn  23903  ufildr  24125  connpconn  35748  cvmliftmolem1  35794  cvmliftmolem2  35795  ordtopconn  36991