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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigaclcu2 | Structured version Visualization version GIF version |
Description: A sigma-algebra is closed under countable union - indexing on ℕ (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
sigaclcu2 | ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun2g 4918 | . . 3 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → ∪ 𝑘 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) | |
2 | 1 | adantl 485 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) |
3 | simpl 486 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
4 | abid 2721 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ↔ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) | |
5 | eleq1a 2829 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) | |
6 | 5 | ralimi 3076 | . . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → ∀𝑘 ∈ ℕ (𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) |
7 | r19.23v 3190 | . . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ (𝑥 = 𝐴 → 𝑥 ∈ 𝑆) ↔ (∃𝑘 ∈ ℕ 𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) | |
8 | 6, 7 | sylib 221 | . . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → (∃𝑘 ∈ ℕ 𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) |
9 | 8 | imp 410 | . . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 ∧ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) → 𝑥 ∈ 𝑆) |
10 | 9 | adantll 714 | . . . . . . 7 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) ∧ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) → 𝑥 ∈ 𝑆) |
11 | 4, 10 | sylan2b 597 | . . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) → 𝑥 ∈ 𝑆) |
12 | 11 | ralrimiva 3097 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}𝑥 ∈ 𝑆) |
13 | nfab1 2902 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} | |
14 | nfcv 2900 | . . . . . 6 ⊢ Ⅎ𝑥𝑆 | |
15 | 13, 14 | dfss3f 3869 | . . . . 5 ⊢ ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆 ↔ ∀𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}𝑥 ∈ 𝑆) |
16 | 12, 15 | sylibr 237 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆) |
17 | elpw2g 5213 | . . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ↔ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆)) | |
18 | 17 | adantr 484 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ↔ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆)) |
19 | 16, 18 | mpbird 260 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆) |
20 | nnct 13443 | . . . 4 ⊢ ℕ ≼ ω | |
21 | abrexct 30629 | . . . 4 ⊢ (ℕ ≼ ω → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) | |
22 | 20, 21 | mp1i 13 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) |
23 | sigaclcu 31658 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ∧ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) → ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝑆) | |
24 | 3, 19, 22, 23 | syl3anc 1372 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝑆) |
25 | 2, 24 | eqeltrd 2834 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {cab 2717 ∀wral 3054 ∃wrex 3055 ⊆ wss 3844 𝒫 cpw 4489 ∪ cuni 4797 ∪ ciun 4882 class class class wbr 5031 ran crn 5527 ωcom 7602 ≼ cdom 8556 ℕcn 11719 sigAlgebracsiga 31649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-card 9444 df-acn 9447 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-n0 11980 df-z 12066 df-uz 12328 df-siga 31650 |
This theorem is referenced by: sigaclfu2 31662 sigaclcu3 31663 measiun 31759 |
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