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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigapisys | Structured version Visualization version GIF version |
Description: All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.) |
Ref | Expression |
---|---|
ispisys.p | ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠} |
Ref | Expression |
---|---|
sigapisys | ⊢ (sigAlgebra‘𝑂) ⊆ 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sigasspw 29506 | . . . . 5 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ⊆ 𝒫 𝑂) | |
2 | selpw 4115 | . . . . 5 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑂 ↔ 𝑡 ⊆ 𝒫 𝑂) | |
3 | 1, 2 | sylibr 223 | . . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝒫 𝒫 𝑂) |
4 | elrnsiga 29516 | . . . . . . 7 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ ∪ ran sigAlgebra) | |
5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑡 ∈ ∪ ran sigAlgebra) |
6 | eldifsn 4260 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑡 ∩ Fin) ∧ 𝑥 ≠ ∅)) | |
7 | 6 | biimpi 205 | . . . . . . . . 9 ⊢ (𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅}) → (𝑥 ∈ (𝒫 𝑡 ∩ Fin) ∧ 𝑥 ≠ ∅)) |
8 | 7 | adantl 481 | . . . . . . . 8 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑡 ∩ Fin) ∧ 𝑥 ≠ ∅)) |
9 | 8 | simpld 474 | . . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑡 ∩ Fin)) |
10 | 9 | elin1d 3764 | . . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑡) |
11 | 9 | elin2d 3765 | . . . . . . 7 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin) |
12 | fict 8433 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω) | |
13 | 11, 12 | syl 17 | . . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑥 ≼ ω) |
14 | 8 | simprd 478 | . . . . . 6 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅) |
15 | sigaclci 29522 | . . . . . 6 ⊢ (((𝑡 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ 𝑡) | |
16 | 5, 10, 13, 14, 15 | syl22anc 1319 | . . . . 5 ⊢ ((𝑡 ∈ (sigAlgebra‘𝑂) ∧ 𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ 𝑡) |
17 | 16 | ralrimiva 2949 | . . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → ∀𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑡) |
18 | 3, 17 | jca 553 | . . 3 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑡)) |
19 | ispisys.p | . . . 4 ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠} | |
20 | 19 | ispisys2 29543 | . . 3 ⊢ (𝑡 ∈ 𝑃 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑡 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑡)) |
21 | 18, 20 | sylibr 223 | . 2 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) → 𝑡 ∈ 𝑃) |
22 | 21 | ssriv 3572 | 1 ⊢ (sigAlgebra‘𝑂) ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ∩ cint 4410 class class class wbr 4583 ran crn 5039 ‘cfv 5804 ωcom 6957 ≼ cdom 7839 Fincfn 7841 ficfi 8199 sigAlgebracsiga 29497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-card 8648 df-acn 8651 df-ac 8822 df-siga 29498 |
This theorem is referenced by: sigapildsys 29552 |
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