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Theorem indisconn 21954
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 21538 . 2 {∅, 𝐴} ∈ Top
2 inss1 4202 . . 3 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3 indislem 21536 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
42, 3sseqtrri 4001 . 2 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5 indisuni 21539 . . 3 ( I ‘𝐴) = {∅, 𝐴}
65isconn2 21950 . 2 ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
71, 4, 6mpbir2an 707 1 {∅, 𝐴} ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  cin 3932  wss 3933  c0 4288  {cpr 4559   I cid 5452  cfv 6348  Topctop 21429  Clsdccld 21552  Conncconn 21947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447  df-cld 21555  df-conn 21948
This theorem is referenced by:  conncompid  21967  cvmlift2lem9  32455
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