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| Mirrors > Home > MPE Home > Th. List > Mathboxes > difelsiga | Structured version Visualization version GIF version | ||
| Description: A sigma-algebra is closed under class differences. (Contributed by Thierry Arnoux, 13-Sep-2016.) |
| Ref | Expression |
|---|---|
| difelsiga | ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp2 1138 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆) | |
| 2 | elssuni 4937 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆) | |
| 3 | difin2 4301 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝑆 → (𝐴 ∖ 𝐵) = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) | |
| 4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) |
| 5 | isrnsigau 34128 | . . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) | |
| 6 | 5 | simprd 495 | . . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) |
| 7 | 6 | simp2d 1144 | . . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆) |
| 8 | difeq2 4120 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ 𝐵)) | |
| 9 | 8 | eleq1d 2826 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) |
| 10 | 9 | rspccva 3621 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
| 11 | 7, 10 | sylan 580 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
| 12 | 11 | 3adant2 1132 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
| 13 | intprg 4981 | . . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) | |
| 14 | 12, 1, 13 | syl2anc 584 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) |
| 15 | 4, 14 | eqtr4d 2780 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴}) |
| 16 | simp1 1137 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
| 17 | prssi 4821 | . . . . 5 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) | |
| 18 | 12, 1, 17 | syl2anc 584 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) |
| 19 | prex 5437 | . . . . 5 ⊢ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ V | |
| 20 | 19 | elpw 4604 | . . . 4 ⊢ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) |
| 21 | 18, 20 | sylibr 234 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆) |
| 22 | prct 32726 | . . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω) | |
| 23 | 12, 1, 22 | syl2anc 584 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω) |
| 24 | prnzg 4778 | . . . 4 ⊢ ((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅) | |
| 25 | 12, 24 | syl 17 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅) |
| 26 | sigaclci 34133 | . . 3 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆) ∧ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω ∧ {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅)) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆) | |
| 27 | 16, 21, 23, 25, 26 | syl22anc 839 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆) |
| 28 | 15, 27 | eqeltrd 2841 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 ∩ cint 4946 class class class wbr 5143 ran crn 5686 ωcom 7887 ≼ cdom 8983 sigAlgebracsiga 34109 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-siga 34110 |
| This theorem is referenced by: inelsiga 34136 sigainb 34137 sigaldsys 34160 cldssbrsiga 34188 measxun2 34211 measssd 34216 measunl 34217 measiuns 34218 measiun 34219 meascnbl 34220 imambfm 34264 dya2iocbrsiga 34277 dya2icobrsiga 34278 sxbrsigalem2 34288 probdif 34422 |
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