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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 20806 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
2 | inss1 3833 | . . 3 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, 𝐴} | |
3 | indislem 20804 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
4 | 2, 3 | sseqtr4i 3638 | . 2 ⊢ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)} |
5 | indisuni 20807 | . . 3 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | |
6 | 5 | isconn2 21217 | . 2 ⊢ ({∅, 𝐴} ∈ Conn ↔ ({∅, 𝐴} ∈ Top ∧ ({∅, 𝐴} ∩ (Clsd‘{∅, 𝐴})) ⊆ {∅, ( I ‘𝐴)})) |
7 | 1, 4, 6 | mpbir2an 955 | 1 ⊢ {∅, 𝐴} ∈ Conn |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {cpr 4179 I cid 5023 ‘cfv 5888 Topctop 20698 Clsdccld 20820 Conncconn 21214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-topon 20716 df-cld 20823 df-conn 21215 |
This theorem is referenced by: conncompid 21234 cvmlift2lem9 31293 |
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