Theorem cldssbrsiga 34200

Description: A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)

Assertion

Ref	Expression
cldssbrsiga	⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

Proof of Theorem cldssbrsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 22944	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
3	2	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
4		dfss4 4216	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
5	3, 4	sylib 218	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
6	1	topopn 22821	. . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
7	1	difopn 22949	. . . . . 6 ⊢ ((∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
8	6, 7	sylan 580	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
9		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
10	9	sgsiga 34155	. . . . . . 7 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	10	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
12		elex 3457	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
13		sigagensiga 34154	. . . . . . . 8 ⊢ (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽))
14		baselsiga 34128	. . . . . . . 8 ⊢ ((sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
15	12, 13, 14	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (sigaGen‘𝐽))
16	15	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
17		elsigagen 34160	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽))
18		difelsiga 34146	. . . . . 6 ⊢ (((sigaGen‘𝐽) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐽 ∈ (sigaGen‘𝐽) ∧ (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
19	11, 16, 17, 18	syl3anc 1373	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
20	8, 19	syldan 591	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
21	5, 20	eqeltrrd 2832	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
22	21	ex 412	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
23	22	ssrdv 3935	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897  ∪ cuni 4856  ran crn 5615  ‘cfv 6481  Topctop 22808  Clsdccld 22931  sigAlgebracsiga 34121  sigaGencsigagen 34151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-ac2 10354

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-dju 9794  df-card 9832  df-acn 9835  df-ac 10007  df-top 22809  df-cld 22934  df-siga 34122  df-sigagen 34152

This theorem is referenced by:  sxbrsigalem4  34300  sibfinima  34352  sibfof  34353  orvccel  34476