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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 22375 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
2 | inss1 4192 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
3 | indislem 22373 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
4 | 2, 3 | sseqtrri 3985 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} |
5 | indisuni 22376 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
6 | 5 | isconn2 22788 | . 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 2107 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 {cpr 4592 I cid 5534 ‘cfv 6500 Topctop 22265 Clsdccld 22390 Conncconn 22785 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-top 22266 df-topon 22283 df-cld 22393 df-conn 22786 |
This theorem is referenced by: conncompid 22805 cvmlift2lem9 33969 |
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