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Mirrors > Home > MPE Home > Th. List > indisconn | Structured version Visualization version GIF version |
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 22336 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
2 | inss1 4186 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
3 | indislem 22334 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
4 | 2, 3 | sseqtrri 3979 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} |
5 | indisuni 22337 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
6 | 5 | isconn2 22749 | . 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)})) |
7 | 1, 4, 6 | mpbir2an 709 | 1 ⊢ {∅, 𝐴} ∈ Conn |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∩ cin 3907 ⊆ wss 3908 ∅c0 4280 {cpr 4586 I cid 5528 ‘cfv 6493 Topctop 22226 Clsdccld 22351 Conncconn 22746 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-top 22227 df-topon 22244 df-cld 22354 df-conn 22747 |
This theorem is referenced by: conncompid 22766 cvmlift2lem9 33774 |
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