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Theorem sigaldsys 31438
Description: All sigma-algebras are lambda-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
Hypothesis
Ref Expression
isldsys.l 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
Assertion
Ref Expression
sigaldsys (sigAlgebra‘𝑂) ⊆ 𝐿
Distinct variable groups:   𝑦,𝑠   𝑂,𝑠,𝑥
Allowed substitution hints:   𝐿(𝑥,𝑦,𝑠)   𝑂(𝑦)

Proof of Theorem sigaldsys
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 sigasspw 31395 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2 velpw 4525 . . . . 5 (𝑡 ∈ 𝒫 𝒫 𝑂𝑡 ⊆ 𝒫 𝑂)
31, 2sylibr 237 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4 elrnsiga 31405 . . . . . 6 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ran sigAlgebra)
5 0elsiga 31393 . . . . . 6 (𝑡 ran sigAlgebra → ∅ ∈ 𝑡)
64, 5syl 17 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
74adantr 484 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑡 ran sigAlgebra)
8 baselsiga 31394 . . . . . . . 8 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂𝑡)
98adantr 484 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑂𝑡)
10 simpr 488 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑥𝑡)
11 difelsiga 31412 . . . . . . 7 ((𝑡 ran sigAlgebra ∧ 𝑂𝑡𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
127, 9, 10, 11syl3anc 1368 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
1312ralrimiva 3176 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
144ad2antrr 725 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑡 ran sigAlgebra)
15 simplr 768 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16 simprl 770 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
17 sigaclcu 31396 . . . . . . . 8 ((𝑡 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡𝑥 ≼ ω) → 𝑥𝑡)
1814, 15, 16, 17syl3anc 1368 . . . . . . 7 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
1918ex 416 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
2019ralrimiva 3176 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
216, 13, 203jca 1125 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
223, 21jca 515 . . 3 (𝑡 ∈ (sigAlgebra‘𝑂) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
23 isldsys.l . . . 4 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
2423isldsys 31435 . . 3 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
2522, 24sylibr 237 . 2 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡𝐿)
2625ssriv 3955 1 (sigAlgebra‘𝑂) ⊆ 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  {crab 3136  cdif 3915  wss 3918  c0 4274  𝒫 cpw 4520   cuni 4821  Disj wdisj 5014   class class class wbr 5049  ran crn 5539  cfv 6338  ωcom 7565  cdom 8492  sigAlgebracsiga 31387
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 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-inf2 9090  ax-ac2 9872
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 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-iin 4905  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-se 5498  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-isom 6347  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-oi 8960  df-dju 9316  df-card 9354  df-acn 9357  df-ac 9529  df-siga 31388
This theorem is referenced by:  ldsysgenld  31439  sigapildsys  31441
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