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Theorem indisconn 21202
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 20787 . 2 {∅, 𝐴} ∈ Top
2 inss1 3825 . . 3 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3 indislem 20785 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
42, 3sseqtr4i 3630 . 2 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5 indisuni 20788 . . 3 ( I ‘𝐴) = {∅, 𝐴}
65isconn2 21198 . 2 ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
71, 4, 6mpbir2an 954 1 {∅, 𝐴} ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  wcel 1988  cin 3566  wss 3567  c0 3907  {cpr 4170   I cid 5013  cfv 5876  Topctop 20679  Clsdccld 20801  Conncconn 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-top 20680  df-topon 20697  df-cld 20804  df-conn 21196
This theorem is referenced by:  conncompid  21215  cvmlift2lem9  31267
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