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| 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.) | 
| Ref | Expression | 
|---|---|
| indisconn | ⊢ {∅, 𝐴} ∈ Conn | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | indistop 23009 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
| 2 | inss1 4237 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
| 3 | indislem 23007 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
| 4 | 2, 3 | sseqtrri 4033 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} | 
| 5 | indisuni 23010 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
| 6 | 5 | isconn2 23422 | . 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)})) | 
| 7 | 1, 4, 6 | mpbir2an 711 | 1 ⊢ {∅, 𝐴} ∈ Conn | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {cpr 4628 I cid 5577 ‘cfv 6561 Topctop 22899 Clsdccld 23024 Conncconn 23419 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-top 22900 df-topon 22917 df-cld 23027 df-conn 23420 | 
| This theorem is referenced by: conncompid 23439 cvmlift2lem9 35316 | 
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