Theorem conntop 23446

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 2740	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23442	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∩ cin 3975  ∅c0 4352  {cpr 4650  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  Clsdccld 23045  Conncconn 23440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-conn 23441

This theorem is referenced by:  conncompss  23462  txconn  23718  qtopconn  23738  ufildr  23960  connpconn  35203  cvmliftmolem1  35249  cvmliftmolem2  35250  ordtopconn  36405