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Mirrors > Home > MPE Home > Th. List > cldmre | Structured version Visualization version GIF version |
Description: The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cldmre | ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | cldss2 21635 | . . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋) |
4 | 1 | topcld 21640 | . 2 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
5 | intcld 21645 | . . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ (Clsd‘𝐽)) | |
6 | 5 | ancoms 462 | . . 3 ⊢ ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽)) |
7 | 6 | 3adant1 1127 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽)) |
8 | 3, 4, 7 | ismred 16865 | 1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 ∩ cint 4838 ‘cfv 6324 Moorecmre 16845 Topctop 21498 Clsdccld 21621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-mre 16849 df-top 21499 df-cld 21624 |
This theorem is referenced by: mrccls 21684 cldmreon 21699 mreclatdemoBAD 21701 |
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