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Theorem cldssbrsiga 31506
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 2824 . . . . . . 7 𝐽 = 𝐽
21cldss 21640 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
32adantl 485 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 𝐽)
4 dfss4 4220 . . . . 5 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
53, 4sylib 221 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
61topopn 21517 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
71difopn 21645 . . . . . 6 (( 𝐽𝐽𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
86, 7sylan 583 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
9 id 22 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ Top)
109sgsiga 31461 . . . . . . 7 (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ran sigAlgebra)
1110adantr 484 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ran sigAlgebra)
12 elex 3498 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ V)
13 sigagensiga 31460 . . . . . . . 8 (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽))
14 baselsiga 31434 . . . . . . . 8 ((sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
1512, 13, 143syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ (sigaGen‘𝐽))
1615adantr 484 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
17 elsigagen 31466 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽𝑥) ∈ (sigaGen‘𝐽))
18 difelsiga 31452 . . . . . 6 (((sigaGen‘𝐽) ∈ ran sigAlgebra ∧ 𝐽 ∈ (sigaGen‘𝐽) ∧ ( 𝐽𝑥) ∈ (sigaGen‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
1911, 16, 17, 18syl3anc 1368 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
208, 19syldan 594 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
215, 20eqeltrrd 2917 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
2221ex 416 . 2 (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
2322ssrdv 3959 1 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cdif 3916  wss 3919   cuni 4824  ran crn 5543  cfv 6343  Topctop 21504  Clsdccld 21627  sigAlgebracsiga 31427  sigaGencsigagen 31457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-ac2 9883
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-oi 8971  df-dju 9327  df-card 9365  df-acn 9368  df-ac 9540  df-top 21505  df-cld 21630  df-siga 31428  df-sigagen 31458
This theorem is referenced by:  sxbrsigalem4  31605  sibfinima  31657  sibfof  31658  orvccel  31780
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