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Theorem sigaldsys 30207
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 30164 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2 selpw 4163 . . . . 5 (𝑡 ∈ 𝒫 𝒫 𝑂𝑡 ⊆ 𝒫 𝑂)
31, 2sylibr 224 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4 elrnsiga 30174 . . . . . 6 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ran sigAlgebra)
5 0elsiga 30162 . . . . . 6 (𝑡 ran sigAlgebra → ∅ ∈ 𝑡)
64, 5syl 17 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
74adantr 481 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑡 ran sigAlgebra)
8 baselsiga 30163 . . . . . . . 8 (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂𝑡)
98adantr 481 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑂𝑡)
10 simpr 477 . . . . . . 7 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → 𝑥𝑡)
11 difelsiga 30181 . . . . . . 7 ((𝑡 ran sigAlgebra ∧ 𝑂𝑡𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
127, 9, 10, 11syl3anc 1325 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
1312ralrimiva 2965 . . . . 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 792 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16 simprl 794 . . . . . . . 8 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
17 sigaclcu 30165 . . . . . . . 8 ((𝑡 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡𝑥 ≼ ω) → 𝑥𝑡)
1814, 15, 16, 17syl3anc 1325 . . . . . . 7 (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
1918ex 450 . . . . . 6 ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
2019ralrimiva 2965 . . . . 5 (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
216, 13, 203jca 1241 . . . 4 (𝑡 ∈ (sigAlgebra‘𝑂) → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
223, 21jca 554 . . 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 30204 . . 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 3605 1 (sigAlgebra‘𝑂) ⊆ 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  {crab 2915  cdif 3569  wss 3572  c0 3913  𝒫 cpw 4156   cuni 4434  Disj wdisj 4618   class class class wbr 4651  ran crn 5113  cfv 5886  ωcom 7062  cdom 7950  sigAlgebracsiga 30155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-ac2 9282
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-oi 8412  df-card 8762  df-acn 8765  df-ac 8936  df-cda 8987  df-siga 30156
This theorem is referenced by:  ldsysgenld  30208  sigapildsys  30210
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