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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 21607 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
2 | inss1 4155 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
3 | indislem 21605 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
4 | 2, 3 | sseqtrri 3952 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} |
5 | indisuni 21608 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
6 | 5 | isconn2 22019 | . 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)})) |
7 | 1, 4, 6 | mpbir2an 710 | 1 ⊢ {∅, 𝐴} ∈ Conn |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {cpr 4527 I cid 5424 ‘cfv 6324 Topctop 21498 Clsdccld 21621 Conncconn 22016 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-topon 21516 df-cld 21624 df-conn 22017 |
This theorem is referenced by: conncompid 22036 cvmlift2lem9 32671 |
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