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Mirrors > Home > MPE Home > Th. List > isconn2 | Structured version Visualization version GIF version |
Description: The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
isconn.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isconn2 | ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isconn.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | isconn 21949 | . 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
3 | 0opn 21440 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
4 | 0cld 21574 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 4168 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | 1 | topopn 21442 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
7 | 1 | topcld 21571 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
8 | 6, 7 | elind 4168 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
9 | 5, 8 | prssd 4747 | . . . . 5 ⊢ (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))) |
10 | 9 | biantrud 532 | . . . 4 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))) |
11 | eqss 3979 | . . . 4 ⊢ ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))) | |
12 | 10, 11 | syl6rbbr 291 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
13 | 12 | pm5.32i 575 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
14 | 2, 13 | bitri 276 | 1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {cpr 4559 ∪ cuni 4830 ‘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 |
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-cld 21555 df-conn 21948 |
This theorem is referenced by: indisconn 21954 dfconn2 21955 cnconn 21958 txconn 22225 filconn 22419 onsucconni 33682 |
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