Theorem conntop 22019

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 2821	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 22015	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 500	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ∩ cin 3934  ∅c0 4290  {cpr 4562  ∪ cuni 4831  ‘cfv 6349  Topctop 21495  Clsdccld 21618  Conncconn 22013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-conn 22014

This theorem is referenced by:  conncompss  22035  txconn  22291  qtopconn  22311  ufildr  22533  connpconn  32477  cvmliftmolem1  32523  cvmliftmolem2  32524  ordtopconn  33782