Theorem sigaldsys 34838

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 34794	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂)
2		velpw 4561	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑂 ↔ 𝑡 ⊆ 𝒫 𝑂)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂)
4		elrnsiga 34804	. . . . . 6 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ ∪ ran sigAlgebra)
5		0elsiga 34792	. . . . . 6 ⊢ (𝑡 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑡)
6	4, 5	syl 18	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∅ ∈ 𝑡)
7	4	adantr 486	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑡 ∈ ∪ ran sigAlgebra)
8		baselsiga 34793	. . . . . . . 8 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ 𝑡)
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑂 ∈ 𝑡)
10		simpr 490	. . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → 𝑥 ∈ 𝑡)
11		difelsiga 34813	. . . . . . 7 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑂 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
12	7, 9, 10, 11	syl3anc 1398	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
13	12	ralrimiva 3154	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
14	4	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑡 ∈ ∪ ran sigAlgebra)
15		simplr 781	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑡)
16		simprl 783	. . . . . . . 8 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
17		sigaclcu 34795	. . . . . . . 8 ⊢ ((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
18	14, 15, 16, 17	syl3anc 1398	. . . . . . 7 ⊢ (((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑡)
19	18	ex 418	. . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
20	19	ralrimiva 3154	. . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
21	6, 13, 20	3jca 1146	. . . 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 521	. . 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 34835	. . 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 3912	1 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  Disj wdisj 5069   class class class wbr 5102  ran crn 5651  ‘cfv 6538  ωcom 7877   ≼ cdom 8974  sigAlgebracsiga 34786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-oi 9507  df-dju 9985  df-card 10023  df-siga 34787

This theorem is used by:  ldsysgenld  34839  sigapildsys  34841