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Theorem difelsiga 30521
Description: A sigma-algebra is closed under class differences. (Contributed by Thierry Arnoux, 13-Sep-2016.)
Assertion
Ref Expression
difelsiga ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)

Proof of Theorem difelsiga
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp2 1160 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 elssuni 4661 . . . 4 (𝐴𝑆𝐴 𝑆)
3 difin2 4091 . . . 4 (𝐴 𝑆 → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
41, 2, 33syl 18 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
5 isrnsigau 30515 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
65simprd 485 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
76simp2d 1166 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
8 difeq2 3921 . . . . . . . 8 (𝑥 = 𝐵 → ( 𝑆𝑥) = ( 𝑆𝐵))
98eleq1d 2870 . . . . . . 7 (𝑥 = 𝐵 → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆𝐵) ∈ 𝑆))
109rspccva 3501 . . . . . 6 ((∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
117, 10sylan 571 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
12113adant2 1154 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
13 intprg 4703 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
1412, 1, 13syl2anc 575 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
154, 14eqtr4d 2843 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = {( 𝑆𝐵), 𝐴})
16 simp1 1159 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝑆 ran sigAlgebra)
17 prssi 4542 . . . . 5 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
1812, 1, 17syl2anc 575 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
19 prex 5099 . . . . 5 {( 𝑆𝐵), 𝐴} ∈ V
2019elpw 4357 . . . 4 ({( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
2118, 20sylibr 225 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆)
22 prct 29819 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
2312, 1, 22syl2anc 575 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
24 prnzg 4501 . . . 4 (( 𝑆𝐵) ∈ 𝑆 → {( 𝑆𝐵), 𝐴} ≠ ∅)
2512, 24syl 17 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≠ ∅)
26 sigaclci 30520 . . 3 (((𝑆 ran sigAlgebra ∧ {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆) ∧ ({( 𝑆𝐵), 𝐴} ≼ ω ∧ {( 𝑆𝐵), 𝐴} ≠ ∅)) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2716, 21, 23, 25, 26syl22anc 858 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2815, 27eqeltrd 2885 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096  cdif 3766  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  {cpr 4372   cuni 4630   cint 4669   class class class wbr 4844  ran crn 5312  ωcom 7295  cdom 8190  sigAlgebracsiga 30495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-ac2 9570
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-oi 8654  df-card 9048  df-acn 9051  df-ac 9222  df-cda 9275  df-siga 30496
This theorem is referenced by:  inelsiga  30523  sigainb  30524  sigaldsys  30547  cldssbrsiga  30575  measxun2  30598  measssd  30603  measunl  30604  measiuns  30605  measiun  30606  meascnbl  30607  imambfm  30649  dya2iocbrsiga  30662  dya2icobrsiga  30663  sxbrsigalem2  30673  probdif  30807
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