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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 21538 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
2 | inss1 4202 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
3 | indislem 21536 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
4 | 2, 3 | sseqtrri 4001 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} |
5 | indisuni 21539 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
6 | 5 | isconn2 21950 | . 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)})) |
7 | 1, 4, 6 | mpbir2an 707 | 1 ⊢ {∅, 𝐴} ∈ Conn |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {cpr 4559 I cid 5452 ‘cfv 6348 Topctop 21429 Clsdccld 21552 Conncconn 21947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-top 21430 df-topon 21447 df-cld 21555 df-conn 21948 |
This theorem is referenced by: conncompid 21967 cvmlift2lem9 32455 |
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