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Theorem sigaldsys 34160
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 34117 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2 velpw 4605 . . . . 5 (𝑡 ∈ 𝒫 𝒫 𝑂𝑡 ⊆ 𝒫 𝑂)
31, 2sylibr 234 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4 elrnsiga 34127 . . . . . 6 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ran sigAlgebra)
5 0elsiga 34115 . . . . . 6 (𝑡 ran sigAlgebra → ∅ ∈ 𝑡)
64, 5syl 17 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
74adantr 480 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑡 ran sigAlgebra)
8 baselsiga 34116 . . . . . . . 8 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂𝑡)
98adantr 480 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑂𝑡)
10 simpr 484 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑥𝑡)
11 difelsiga 34134 . . . . . . 7 ((𝑡 ran sigAlgebra ∧ 𝑂𝑡𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
127, 9, 10, 11syl3anc 1373 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
1312ralrimiva 3146 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
144ad2antrr 726 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑡 ran sigAlgebra)
15 simplr 769 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16 simprl 771 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
17 sigaclcu 34118 . . . . . . . 8 ((𝑡 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡𝑥 ≼ ω) → 𝑥𝑡)
1814, 15, 16, 17syl3anc 1373 . . . . . . 7 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
1918ex 412 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
2019ralrimiva 3146 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
216, 13, 203jca 1129 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
223, 21jca 511 . . 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 34157 . . 3 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
2522, 24sylibr 234 . 2 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡𝐿)
2625ssriv 3987 1 (sigAlgebra‘𝑂) ⊆ 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907  Disj wdisj 5110   class class class wbr 5143  ran crn 5686  cfv 6561  ωcom 7887  cdom 8983  sigAlgebracsiga 34109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-siga 34110
This theorem is referenced by:  ldsysgenld  34161  sigapildsys  34163
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