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Theorem conntop 23441
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
StepHypRef Expression
1 eqid 2735 . . 3 𝐽 = 𝐽
21isconn 23437 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝐽}))
32simplbi 497 1 (𝐽 ∈ Conn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cin 3962  c0 4339  {cpr 4633   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  Conncconn 23435
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-conn 23436
This theorem is referenced by:  conncompss  23457  txconn  23713  qtopconn  23733  ufildr  23955  connpconn  35220  cvmliftmolem1  35266  cvmliftmolem2  35267  ordtopconn  36422
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