Theorem indisconn 22569

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 22152	. 2 ⊢ {∅, 𝐴} ∈ Top
2		inss1 4162	. . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴}
3		indislem 22150	. . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
4	2, 3	sseqtrri 3958	. 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}
5		indisuni 22153	. . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
6	5	isconn2 22565	. 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)}))
7	1, 4, 6	mpbir2an 708	1 ⊢ {∅, 𝐴} ∈ Conn

Syntax hints:   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {cpr 4563   I cid 5488  ‘cfv 6433  Topctop 22042  Clsdccld 22167  Conncconn 22562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-top 22043  df-topon 22060  df-cld 22170  df-conn 22563

This theorem is referenced by:  conncompid  22582  cvmlift2lem9  33273