Theorem unicls 34646

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

Step	Hyp	Ref	Expression
1		unicls.2	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	cldss2 23403	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3		sspwuni 5065	. . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋)
4	2, 3	mpbi 233	. 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋
5		unicls.1	. . 3 ⊢ 𝐽 ∈ Top
6	1	topcld 23408	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
7	5, 6	ax-mp 5	. 2 ⊢ 𝑋 ∈ (Clsd‘𝐽)
8		unissel 4905	. 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋)
9	4, 7, 8	mp2an 705	1 ⊢ ∪ (Clsd‘𝐽) = 𝑋

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4872  ‘cfv 6566  Topctop 23266  Clsdccld 23389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574  df-top 23267  df-cld 23392

This theorem is used by:  sxbrsigalem3  35015  sxbrsigalem4  35030