Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  indiscon Structured version   Visualization version   GIF version

Theorem indiscon 21031
 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
indiscon {∅, 𝐴} ∈ Con

Proof of Theorem indiscon
StepHypRef Expression
1 indistop 20616 . 2 {∅, 𝐴} ∈ Top
2 inss1 3795 . . 3 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3 indislem 20614 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
42, 3sseqtr4i 3601 . 2 ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5 indisuni 20617 . . 3 ( I ‘𝐴) = {∅, 𝐴}
65iscon2 21027 . 2 ({∅, 𝐴} ∈ Con ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
71, 4, 6mpbir2an 957 1 {∅, 𝐴} ∈ Con
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127   I cid 4948  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523  df-cld 20633  df-con 21025 This theorem is referenced by:  concompid  21044  cvmlift2lem9  30547
 Copyright terms: Public domain W3C validator