MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldmre Structured version   Visualization version   GIF version

Theorem cldmre 20930
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 20882 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
32a1i 11 . 2 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
41topcld 20887 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5 intcld 20892 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
65ancoms 468 . . 3 ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
763adant1 1099 . 2 ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Clsd‘𝐽))
83, 4, 7ismred 16309 1 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wne 2823  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507  cfv 5926  Moorecmre 16289  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-mre 16293  df-top 20747  df-cld 20871
This theorem is referenced by:  mrccls  20931  cldmreon  20946  mreclatdemoBAD  20948
  Copyright terms: Public domain W3C validator