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Theorem sigaldsys 30350
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 30307 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2 selpw 4198 . . . . 5 (𝑡 ∈ 𝒫 𝒫 𝑂𝑡 ⊆ 𝒫 𝑂)
31, 2sylibr 224 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4 elrnsiga 30317 . . . . . 6 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ran sigAlgebra)
5 0elsiga 30305 . . . . . 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 30306 . . . . . . . 8 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂𝑡)
98adantr 480 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑂𝑡)
10 simpr 476 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑥𝑡)
11 difelsiga 30324 . . . . . . 7 ((𝑡 ran sigAlgebra ∧ 𝑂𝑡𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
127, 9, 10, 11syl3anc 1366 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
1312ralrimiva 2995 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
144ad2antrr 762 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑡 ran sigAlgebra)
15 simplr 807 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16 simprl 809 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
17 sigaclcu 30308 . . . . . . . 8 ((𝑡 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡𝑥 ≼ ω) → 𝑥𝑡)
1814, 15, 16, 17syl3anc 1366 . . . . . . 7 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
1918ex 449 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
2019ralrimiva 2995 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
216, 13, 203jca 1261 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
223, 21jca 553 . . 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 30347 . . 3 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
2522, 24sylibr 224 . 2 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡𝐿)
2625ssriv 3640 1 (sigAlgebra‘𝑂) ⊆ 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468  Disj wdisj 4652   class class class wbr 4685  ran crn 5144  cfv 5926  ωcom 7107  cdom 7995  sigAlgebracsiga 30298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-cda 9028  df-siga 30299
This theorem is referenced by:  ldsysgenld  30351  sigapildsys  30353
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