Theorem sigaldsys 34125

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 34082	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2		velpw 4558	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑂 ↔ 𝑡 ⊆ 𝒫 𝑂)
3	1, 2	sylibr 234	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4		elrnsiga 34092	. . . . . 6 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ ∪ ran sigAlgebra)
5		0elsiga 34080	. . . . . 6 ⊢ (𝑡 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑡)
6	4, 5	syl 17	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
7	4	adantr 480	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑡 ∈ ∪ ran sigAlgebra)
8		baselsiga 34081	. . . . . . . 8 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ 𝑡)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑂 ∈ 𝑡)
10		simpr 484	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑥 ∈ 𝑡)
11		difelsiga 34099	. . . . . . 7 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑂 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
12	7, 9, 10, 11	syl3anc 1373	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
13	12	ralrimiva 3121	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
14	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑡 ∈ ∪ ran sigAlgebra)
15		simplr 768	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16		simprl 770	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
17		sigaclcu 34083	. . . . . . . 8 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
18	14, 15, 16, 17	syl3anc 1373	. . . . . . 7 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑡)
19	18	ex 412	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
20	19	ralrimiva 3121	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
21	6, 13, 20	3jca 1128	. . . 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 511	. . 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 34122	. . 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 234	. 2 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝐿)
26	25	ssriv 3941	1 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  ∪ cuni 4861  Disj wdisj 5062   class class class wbr 5095  ran crn 5624  ‘cfv 6486  ωcom 7806   ≼ cdom 8877  sigAlgebracsiga 34074

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-dju 9816  df-card 9854  df-acn 9857  df-ac 10029  df-siga 34075

This theorem is referenced by:  ldsysgenld  34126  sigapildsys  34128