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Theorem difelsiga 31385
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 1131 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 elssuni 4859 . . . 4 (𝐴𝑆𝐴 𝑆)
3 difin2 4264 . . . 4 (𝐴 𝑆 → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
41, 2, 33syl 18 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = (( 𝑆𝐵) ∩ 𝐴))
5 isrnsigau 31379 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
65simprd 498 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
76simp2d 1137 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
8 difeq2 4091 . . . . . . . 8 (𝑥 = 𝐵 → ( 𝑆𝑥) = ( 𝑆𝐵))
98eleq1d 2895 . . . . . . 7 (𝑥 = 𝐵 → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆𝐵) ∈ 𝑆))
109rspccva 3620 . . . . . 6 ((∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
117, 10sylan 582 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
12113adant2 1125 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆𝐵) ∈ 𝑆)
13 intprg 4901 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
1412, 1, 13syl2anc 586 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} = (( 𝑆𝐵) ∩ 𝐴))
154, 14eqtr4d 2857 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) = {( 𝑆𝐵), 𝐴})
16 simp1 1130 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝑆 ran sigAlgebra)
17 prssi 4746 . . . . 5 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
1812, 1, 17syl2anc 586 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
19 prex 5323 . . . . 5 {( 𝑆𝐵), 𝐴} ∈ V
2019elpw 4544 . . . 4 ({( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {( 𝑆𝐵), 𝐴} ⊆ 𝑆)
2118, 20sylibr 236 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆)
22 prct 30442 . . . 4 ((( 𝑆𝐵) ∈ 𝑆𝐴𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
2312, 1, 22syl2anc 586 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≼ ω)
24 prnzg 4705 . . . 4 (( 𝑆𝐵) ∈ 𝑆 → {( 𝑆𝐵), 𝐴} ≠ ∅)
2512, 24syl 17 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ≠ ∅)
26 sigaclci 31384 . . 3 (((𝑆 ran sigAlgebra ∧ {( 𝑆𝐵), 𝐴} ∈ 𝒫 𝑆) ∧ ({( 𝑆𝐵), 𝐴} ≼ ω ∧ {( 𝑆𝐵), 𝐴} ≠ ∅)) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2716, 21, 23, 25, 26syl22anc 836 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {( 𝑆𝐵), 𝐴} ∈ 𝑆)
2815, 27eqeltrd 2911 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wne 3014  wral 3136  cdif 3931  cin 3933  wss 3934  c0 4289  𝒫 cpw 4537  {cpr 4561   cuni 4830   cint 4867   class class class wbr 5057  ran crn 5549  ωcom 7572  cdom 8499  sigAlgebracsiga 31360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-ac2 9877
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-ac 9534  df-siga 31361
This theorem is referenced by:  inelsiga  31387  sigainb  31388  sigaldsys  31411  cldssbrsiga  31439  measxun2  31462  measssd  31467  measunl  31468  measiuns  31469  measiun  31470  meascnbl  31471  imambfm  31513  dya2iocbrsiga  31526  dya2icobrsiga  31527  sxbrsigalem2  31537  probdif  31671
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