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Theorem sigaldsys 34140
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 34097 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2 velpw 4610 . . . . 5 (𝑡 ∈ 𝒫 𝒫 𝑂𝑡 ⊆ 𝒫 𝑂)
31, 2sylibr 234 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4 elrnsiga 34107 . . . . . 6 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ran sigAlgebra)
5 0elsiga 34095 . . . . . 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 34096 . . . . . . . 8 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂𝑡)
98adantr 480 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑂𝑡)
10 simpr 484 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑥𝑡)
11 difelsiga 34114 . . . . . . 7 ((𝑡 ran sigAlgebra ∧ 𝑂𝑡𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
127, 9, 10, 11syl3anc 1370 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
1312ralrimiva 3144 . . . . 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 34098 . . . . . . . 8 ((𝑡 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡𝑥 ≼ ω) → 𝑥𝑡)
1814, 15, 16, 17syl3anc 1370 . . . . . . 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 3144 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
216, 13, 203jca 1127 . . . 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 34137 . . 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 3999 1 (sigAlgebra‘𝑂) ⊆ 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  Disj wdisj 5115   class class class wbr 5148  ran crn 5690  cfv 6563  ωcom 7887  cdom 8982  sigAlgebracsiga 34089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-siga 34090
This theorem is referenced by:  ldsysgenld  34141  sigapildsys  34143
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