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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 21632 | . . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋) |
4 | 1 | topcld 21637 | . 2 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
5 | intcld 21642 | . . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ (Clsd‘𝐽)) | |
6 | 5 | ancoms 461 | . . 3 ⊢ ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽)) |
7 | 6 | 3adant1 1126 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽)) |
8 | 3, 4, 7 | ismred 16867 | 1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4832 ∩ cint 4869 ‘cfv 6350 Moorecmre 16847 Topctop 21495 Clsdccld 21618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fn 6353 df-fv 6358 df-mre 16851 df-top 21496 df-cld 21621 |
This theorem is referenced by: mrccls 21681 cldmreon 21696 mreclatdemoBAD 21698 |
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