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Theorem unicls 31046
Description: The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)
Hypotheses
Ref Expression
unicls.1 𝐽 ∈ Top
unicls.2 𝑋 = 𝐽
Assertion
Ref Expression
unicls (Clsd‘𝐽) = 𝑋

Proof of Theorem unicls
StepHypRef Expression
1 unicls.2 . . . 4 𝑋 = 𝐽
21cldss2 21568 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
3 sspwuni 5014 . . 3 ((Clsd‘𝐽) ⊆ 𝒫 𝑋 (Clsd‘𝐽) ⊆ 𝑋)
42, 3mpbi 231 . 2 (Clsd‘𝐽) ⊆ 𝑋
5 unicls.1 . . 3 𝐽 ∈ Top
61topcld 21573 . . 3 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
75, 6ax-mp 5 . 2 𝑋 ∈ (Clsd‘𝐽)
8 unissel 4862 . 2 (( (Clsd‘𝐽) ⊆ 𝑋𝑋 ∈ (Clsd‘𝐽)) → (Clsd‘𝐽) = 𝑋)
94, 7, 8mp2an 688 1 (Clsd‘𝐽) = 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  wss 3935  𝒫 cpw 4537   cuni 4832  cfv 6349  Topctop 21431  Clsdccld 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357  df-top 21432  df-cld 21557
This theorem is referenced by:  sxbrsigalem3  31430  sxbrsigalem4  31445
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