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Theorem unicls 33902
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 23038 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
3 sspwuni 5100 . . 3 ((Clsd‘𝐽) ⊆ 𝒫 𝑋 (Clsd‘𝐽) ⊆ 𝑋)
42, 3mpbi 230 . 2 (Clsd‘𝐽) ⊆ 𝑋
5 unicls.1 . . 3 𝐽 ∈ Top
61topcld 23043 . . 3 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
75, 6ax-mp 5 . 2 𝑋 ∈ (Clsd‘𝐽)
8 unissel 4938 . 2 (( (Clsd‘𝐽) ⊆ 𝑋𝑋 ∈ (Clsd‘𝐽)) → (Clsd‘𝐽) = 𝑋)
94, 7, 8mp2an 692 1 (Clsd‘𝐽) = 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by:  sxbrsigalem3  34274  sxbrsigalem4  34289
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