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Theorem indisconn 22753
Description: The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indisconn {∅, 𝐴} ∈ Conn

Proof of Theorem indisconn
StepHypRef Expression
1 indistop 22336 . 2 {∅, 𝐴} ∈ Top
2 inss1 4186 . . 3 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3 indislem 22334 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
42, 3sseqtrri 3979 . 2 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5 indisuni 22337 . . 3 ( I ‘𝐴) = {∅, 𝐴}
65isconn2 22749 . 2 ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
71, 4, 6mpbir2an 709 1 {∅, 𝐴} ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  cin 3907  wss 3908  c0 4280  {cpr 4586   I cid 5528  cfv 6493  Topctop 22226  Clsdccld 22351  Conncconn 22746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-top 22227  df-topon 22244  df-cld 22354  df-conn 22747
This theorem is referenced by:  conncompid  22766  cvmlift2lem9  33774
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