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Theorem cldmre 21662
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.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldmre (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))

Proof of Theorem cldmre
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 clscld.1 . . . 4 𝑋 = 𝐽
21cldss2 21614 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
32a1i 11 . 2 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
41topcld 21619 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5 intcld 21624 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
65ancoms 461 . . 3 ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
763adant1 1126 . 2 ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
83, 4, 7ismred 16852 1 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wne 3006  wss 3913  c0 4269  𝒫 cpw 4515   cuni 4814   cint 4852  cfv 6331  Moorecmre 16832  Topctop 21477  Clsdccld 21600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fn 6334  df-fv 6339  df-mre 16836  df-top 21478  df-cld 21603
This theorem is referenced by:  mrccls  21663  cldmreon  21678  mreclatdemoBAD  21680
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