![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 1137 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆) | |
2 | elssuni 4941 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆) | |
3 | difin2 4291 | . . . 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 33194 | . . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) | |
6 | 5 | simprd 496 | . . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) |
7 | 6 | simp2d 1143 | . . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆) |
8 | difeq2 4116 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ 𝐵)) | |
9 | 8 | eleq1d 2818 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) |
10 | 9 | rspccva 3611 | . . . . . 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 1131 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
13 | intprg 4985 | . . . 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 2775 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴}) |
16 | simp1 1136 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
17 | prssi 4824 | . . . . 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 5432 | . . . . 5 ⊢ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ V | |
20 | 19 | elpw 4606 | . . . 4 ⊢ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) |
21 | 18, 20 | sylibr 233 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆) |
22 | prct 31977 | . . . 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 4782 | . . . 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 33199 | . . 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 837 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆) |
28 | 15, 27 | eqeltrd 2833 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {cpr 4630 ∪ cuni 4908 ∩ cint 4950 class class class wbr 5148 ran crn 5677 ωcom 7857 ≼ cdom 8939 sigAlgebracsiga 33175 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-ac2 10460 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-siga 33176 |
This theorem is referenced by: inelsiga 33202 sigainb 33203 sigaldsys 33226 cldssbrsiga 33254 measxun2 33277 measssd 33282 measunl 33283 measiuns 33284 measiun 33285 meascnbl 33286 imambfm 33330 dya2iocbrsiga 33343 dya2icobrsiga 33344 sxbrsigalem2 33354 probdif 33488 |
Copyright terms: Public domain | W3C validator |