Theorem sigaldsys 34466

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 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}

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.

Step	Hyp	Ref	Expression
1		sigasspw 34423	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2		velpw 4563	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑂 ↔ 𝑡 ⊆ 𝒫 𝑂)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4		elrnsiga 34433	. . . . . 6 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ ∪ ran sigAlgebra)
5		0elsiga 34421	. . . . . 6 ⊢ (𝑡 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑡)
6	4, 5	syl 18	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
7	4	adantr 485	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑡 ∈ ∪ ran sigAlgebra)
8		baselsiga 34422	. . . . . . . 8 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ 𝑡)
9	8	adantr 485	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑂 ∈ 𝑡)
10		simpr 489	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑥 ∈ 𝑡)
11		difelsiga 34440	. . . . . . 7 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑂 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
12	7, 9, 10, 11	syl3anc 1394	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
13	12	ralrimiva 3157	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
14	4	ad2antrr 738	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑡 ∈ ∪ ran sigAlgebra)
15		simplr 780	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16		simprl 782	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
17		sigaclcu 34424	. . . . . . . 8 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
18	14, 15, 16, 17	syl3anc 1394	. . . . . . 7 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑡)
19	18	ex 417	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
20	19	ralrimiva 3157	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
21	6, 13, 20	3jca 1144	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)))
22	3, 21	jca 520	. . 3 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))))
23		isldsys.l	. . . 4 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
24	23	isldsys 34463	. . 3 ⊢ (𝑡 ∈ 𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))))
25	22, 24	sylibr 237	. 2 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝐿)
26	25	ssriv 3943	1 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  ∪ cuni 4868  Disj wdisj 5072   class class class wbr 5105  ran crn 5653  ‘cfv 6525  ωcom 7850   ≼ cdom 8929  sigAlgebracsiga 34415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-ac2 10435

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-oi 9460  df-dju 9875  df-card 9913  df-acn 9916  df-ac 10088  df-siga 34416

This theorem is referenced by:  ldsysgenld  34467  sigapildsys  34469