Theorem indisconn 23379

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

Step	Hyp	Ref	Expression
1		indistop 22963	. 2 ⊢ {∅, 𝐴} ∈ Top
2		inss1 4191	. . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3		indislem 22961	. . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
4	2, 3	sseqtrri 3985	. 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5		indisuni 22964	. . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
6	5	isconn2 23375	. 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
7	1, 4, 6	mpbir2an 712	1 ⊢ {∅, 𝐴} ∈ Conn

Syntax hints:   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {cpr 4584   I cid 5528  ‘cfv 6502  Topctop 22854  Clsdccld 22977  Conncconn 23372

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6458  df-fun 6504  df-fv 6510  df-top 22855  df-topon 22872  df-cld 22980  df-conn 23373

This theorem is referenced by:  conncompid  23392  cvmlift2lem9  35533