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Theorem cldssbrsiga 30224
 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.
StepHypRef Expression
1 eqid 2620 . . . . . . 7 𝐽 = 𝐽
21cldss 20814 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
32adantl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 𝐽)
4 dfss4 3850 . . . . 5 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
53, 4sylib 208 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
61topopn 20692 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
71difopn 20819 . . . . . 6 (( 𝐽𝐽𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
86, 7sylan 488 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
9 id 22 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ Top)
109sgsiga 30179 . . . . . . 7 (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ran sigAlgebra)
1110adantr 481 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ran sigAlgebra)
12 elex 3207 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ V)
13 sigagensiga 30178 . . . . . . . 8 (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽))
14 baselsiga 30152 . . . . . . . 8 ((sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
1512, 13, 143syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ (sigaGen‘𝐽))
1615adantr 481 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
17 elsigagen 30184 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽𝑥) ∈ (sigaGen‘𝐽))
18 difelsiga 30170 . . . . . 6 (((sigaGen‘𝐽) ∈ ran sigAlgebra ∧ 𝐽 ∈ (sigaGen‘𝐽) ∧ ( 𝐽𝑥) ∈ (sigaGen‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
1911, 16, 17, 18syl3anc 1324 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
208, 19syldan 487 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
215, 20eqeltrrd 2700 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
2221ex 450 . 2 (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
2322ssrdv 3601 1 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195   ∖ cdif 3564   ⊆ wss 3567  ∪ cuni 4427  ran crn 5105  ‘cfv 5876  Topctop 20679  Clsdccld 20801  sigAlgebracsiga 30144  sigaGencsigagen 30175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-cda 8975  df-top 20680  df-cld 20804  df-siga 30145  df-sigagen 30176 This theorem is referenced by:  sxbrsigalem4  30323  sibfinima  30375  sibfof  30376  orvccel  30498
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