Theorem sigaldsys 31528

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 31485	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2		velpw 4502	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑂 ↔ 𝑡 ⊆ 𝒫 𝑂)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4		elrnsiga 31495	. . . . . 6 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ ∪ ran sigAlgebra)
5		0elsiga 31483	. . . . . 6 ⊢ (𝑡 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑡)
6	4, 5	syl 17	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
7	4	adantr 484	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑡 ∈ ∪ ran sigAlgebra)
8		baselsiga 31484	. . . . . . . 8 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ 𝑡)
9	8	adantr 484	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑂 ∈ 𝑡)
10		simpr 488	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑥 ∈ 𝑡)
11		difelsiga 31502	. . . . . . 7 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑂 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
12	7, 9, 10, 11	syl3anc 1368	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
13	12	ralrimiva 3149	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
14	4	ad2antrr 725	. . . . . . . 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 31486	. . . . . . . 8 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
18	14, 15, 16, 17	syl3anc 1368	. . . . . . 7 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑡)
19	18	ex 416	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
20	19	ralrimiva 3149	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
21	6, 13, 20	3jca 1125	. . . 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 515	. . 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 31525	. . 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 3919	1 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800  Disj wdisj 4995   class class class wbr 5030  ran crn 5520  ‘cfv 6324  ωcom 7560   ≼ cdom 8490  sigAlgebracsiga 31477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-ac2 9874

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-dju 9314  df-card 9352  df-acn 9355  df-ac 9527  df-siga 31478

This theorem is referenced by:  ldsysgenld  31529  sigapildsys  31531