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Theorem difelsiga 34678
Description: A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34676 and unelsiga 34677 rather than countable intersection, and so does not use ax-ac 10486. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Proof shortened by Vincent Gonzalez, 17-Aug-2026.)
Assertion
Ref Expression
difelsiga ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)

Proof of Theorem difelsiga
StepHypRef Expression
1 difun1 4245 . . . 4 ( 𝑆 ∖ (( 𝑆𝐴) ∪ 𝐵)) = (( 𝑆 ∖ ( 𝑆𝐴)) ∖ 𝐵)
21a1i 11 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆 ∖ (( 𝑆𝐴) ∪ 𝐵)) = (( 𝑆 ∖ ( 𝑆𝐴)) ∖ 𝐵))
3 simp1 1154 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝑆 ran sigAlgebra)
4 simp2 1155 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐴𝑆)
5 elsigass 34668 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 𝑆)
63, 4, 5syl2anc 596 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐴 𝑆)
7 dfss4 4215 . . . . 5 (𝐴 𝑆 ↔ ( 𝑆 ∖ ( 𝑆𝐴)) = 𝐴)
86, 7sylib 221 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆 ∖ ( 𝑆𝐴)) = 𝐴)
98difeq1d 4073 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (( 𝑆 ∖ ( 𝑆𝐴)) ∖ 𝐵) = (𝐴𝐵))
102, 9eqtrd 2795 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆 ∖ (( 𝑆𝐴) ∪ 𝐵)) = (𝐴𝐵))
11 difunielsiga 34676 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ( 𝑆𝐴) ∈ 𝑆)
123, 4, 11syl2anc 596 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆𝐴) ∈ 𝑆)
13 simp3 1156 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → 𝐵𝑆)
14 unelsiga 34677 . . . 4 ((𝑆 ran sigAlgebra ∧ ( 𝑆𝐴) ∈ 𝑆𝐵𝑆) → (( 𝑆𝐴) ∪ 𝐵) ∈ 𝑆)
153, 12, 13, 14syl3anc 1398 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (( 𝑆𝐴) ∪ 𝐵) ∈ 𝑆)
16 difunielsiga 34676 . . 3 ((𝑆 ran sigAlgebra ∧ (( 𝑆𝐴) ∪ 𝐵) ∈ 𝑆) → ( 𝑆 ∖ (( 𝑆𝐴) ∪ 𝐵)) ∈ 𝑆)
173, 15, 16syl2anc 596 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ( 𝑆 ∖ (( 𝑆𝐴) ∪ 𝐵)) ∈ 𝑆)
1810, 17eqeltrrd 2861 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  w3a 1103   = wceq 1570  wcel 2145  cdif 3896  cun 3897  wss 3899   cuni 4867  ran crn 5656  sigAlgebracsiga 34651
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-oi 9489  df-dju 9931  df-card 9969  df-siga 34652
This theorem is used by:  inelsiga  34679  sigainb  34680  sigaldsys  34703  cldssbrsiga  34731  measxun2  34754  measssd  34759  measunl  34760  measiuns  34761  measiun  34762  meascnbl  34763  imambfm  34806  dya2iocbrsiga  34819  dya2icobrsiga  34820  sxbrsigalem2  34830  probdif  34964
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