Theorem indisconn 23303

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 22887	. 2 ⊢ {∅, 𝐴} ∈ Top
2		inss1 4188	. . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3		indislem 22885	. . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
4	2, 3	sseqtrri 3985	. 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5		indisuni 22888	. . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
6	5	isconn2 23299	. 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
7	1, 4, 6	mpbir2an 711	1 ⊢ {∅, 𝐴} ∈ Conn

Syntax hints:   ∈ wcel 2109   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {cpr 4579   I cid 5513  ‘cfv 6482  Topctop 22778  Clsdccld 22901  Conncconn 23296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-top 22779  df-topon 22796  df-cld 22904  df-conn 23297

This theorem is referenced by:  conncompid  23316  cvmlift2lem9  35294