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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 1133 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆) | |
2 | elssuni 4870 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆) | |
3 | difin2 4268 | . . . 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 31388 | . . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) | |
6 | 5 | simprd 498 | . . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) |
7 | 6 | simp2d 1139 | . . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆) |
8 | difeq2 4095 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ 𝐵)) | |
9 | 8 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) |
10 | 9 | rspccva 3624 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
11 | 7, 10 | sylan 582 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
12 | 11 | 3adant2 1127 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆) |
13 | intprg 4912 | . . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) | |
14 | 12, 1, 13 | syl2anc 586 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴)) |
15 | 4, 14 | eqtr4d 2861 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴}) |
16 | simp1 1132 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
17 | prssi 4756 | . . . . 5 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) | |
18 | 12, 1, 17 | syl2anc 586 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) |
19 | prex 5335 | . . . . 5 ⊢ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ V | |
20 | 19 | elpw 4545 | . . . 4 ⊢ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆) |
21 | 18, 20 | sylibr 236 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆) |
22 | prct 30452 | . . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω) | |
23 | 12, 1, 22 | syl2anc 586 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω) |
24 | prnzg 4715 | . . . 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 31393 | . . 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 836 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆) |
28 | 15, 27 | eqeltrd 2915 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {cpr 4571 ∪ cuni 4840 ∩ cint 4878 class class class wbr 5068 ran crn 5558 ωcom 7582 ≼ cdom 8509 sigAlgebracsiga 31369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-siga 31370 |
This theorem is referenced by: inelsiga 31396 sigainb 31397 sigaldsys 31420 cldssbrsiga 31448 measxun2 31471 measssd 31476 measunl 31477 measiuns 31478 measiun 31479 meascnbl 31480 imambfm 31522 dya2iocbrsiga 31535 dya2icobrsiga 31536 sxbrsigalem2 31546 probdif 31680 |
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