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Theorem difelsiga 33200
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 1137 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 elssuni 4941 . . . 4 (𝐴𝑆𝐴 𝑆)
3 difin2 4291 . . . 4 (𝐴 𝑆 → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
41, 2, 33syl 18 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
5 isrnsigau 33194 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
65simprd 496 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
76simp2d 1143 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
8 difeq2 4116 . . . . . . . 8 (𝑥 = 𝐵 → ( 𝑆𝑥) = ( 𝑆𝐵))
98eleq1d 2818 . . . . . . 7 (𝑥 = 𝐵 → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆𝐵) ∈ 𝑆))
109rspccva 3611 . . . . . 6 ((∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
117, 10sylan 580 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
12113adant2 1131 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
13 intprg 4985 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
1412, 1, 13syl2anc 584 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
154, 14eqtr4d 2775 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = {( 𝑆𝐵), 𝐴})
16 simp1 1136 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝑆 ran sigAlgebra)
17 prssi 4824 . . . . 5 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
1812, 1, 17syl2anc 584 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
19 prex 5432 . . . . 5 {( 𝑆𝐵), 𝐴} ∈ V
2019elpw 4606 . . . 4 ({( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
2118, 20sylibr 233 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆)
22 prct 31977 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
2312, 1, 22syl2anc 584 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
24 prnzg 4782 . . . 4 (( 𝑆𝐵) ∈ 𝑆 → {( 𝑆𝐵), 𝐴} ≠ ∅)
2512, 24syl 17 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≠ ∅)
26 sigaclci 33199 . . 3 (((𝑆 ran sigAlgebra ∧ {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆) ∧ ({( 𝑆𝐵), 𝐴} ≼ ω ∧ {( 𝑆𝐵), 𝐴} ≠ ∅)) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2716, 21, 23, 25, 26syl22anc 837 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2815, 27eqeltrd 2833 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 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
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