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Mirrors > Home > MPE Home > Th. List > iscon2 | Structured version Visualization version GIF version |
Description: The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
iscon.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscon2 | ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscon.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | iscon 21026 | . 2 ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
3 | 0opn 20534 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
4 | 0cld 20652 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 3760 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | 1 | topopn 20536 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
7 | 1 | topcld 20649 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
8 | 6, 7 | elind 3760 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
9 | prssi 4293 | . . . . . 6 ⊢ ((∅ ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))) | |
10 | 5, 8, 9 | syl2anc 691 | . . . . 5 ⊢ (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))) |
11 | 10 | biantrud 527 | . . . 4 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))) |
12 | eqss 3583 | . . . 4 ⊢ ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))) | |
13 | 11, 12 | syl6rbbr 278 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
14 | 13 | pm5.32i 667 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
15 | 2, 14 | bitri 263 | 1 ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 Conccon 21024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-cld 20633 df-con 21025 |
This theorem is referenced by: indiscon 21031 dfcon2 21032 cnconn 21035 txcon 21302 filcon 21497 onsucconi 31606 |
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