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Theorem indisconn 22001
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 21585 . 2 {∅, 𝐴} ∈ Top
2 inss1 4180 . . 3 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3 indislem 21583 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
42, 3sseqtrri 3980 . 2 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5 indisuni 21586 . . 3 ( I ‘𝐴) = {∅, 𝐴}
65isconn2 21997 . 2 ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
71, 4, 6mpbir2an 710 1 {∅, 𝐴} ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  cin 3909  wss 3910  c0 4266  {cpr 4542   I cid 5432  cfv 6328  Topctop 21476  Clsdccld 21599  Conncconn 21994
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-top 21477  df-topon 21494  df-cld 21602  df-conn 21995
This theorem is referenced by:  conncompid  22014  cvmlift2lem9  32565
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