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Theorem cldmre 21103
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 21055 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
32a1i 11 . 2 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
41topcld 21060 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5 intcld 21065 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
65ancoms 446 . . 3 ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
763adant1 1124 . 2 ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
83, 4, 7ismred 16470 1 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  wss 3723  c0 4063  𝒫 cpw 4298   cuni 4575   cint 4612  cfv 6030  Moorecmre 16450  Topctop 20918  Clsdccld 21041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-mre 16454  df-top 20919  df-cld 21044
This theorem is referenced by:  mrccls  21104  cldmreon  21119  mreclatdemoBAD  21121
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