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 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)

Proof of Theorem difelsiga

Step	Hyp	Ref	Expression
1		difun1 4245	. . . 4 ⊢ (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) = ((∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) ∖ 𝐵)
2	1	a1i 11	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) = ((∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) ∖ 𝐵))
3		simp1 1154	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
4		simp2 1155	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
5		elsigass 34668	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
6	3, 4, 5	syl2anc 596	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
7		dfss4 4215	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) = 𝐴)
8	6, 7	sylib 221	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) = 𝐴)
9	8	difeq1d 4073	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) ∖ 𝐵) = (𝐴 ∖ 𝐵))
10	2, 9	eqtrd 2795	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) = (𝐴 ∖ 𝐵))
11		difunielsiga 34676	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)
12	3, 4, 11	syl2anc 596	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)
13		simp3 1156	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
14		unelsiga 34677	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (∪ 𝑆 ∖ 𝐴) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵) ∈ 𝑆)
15	3, 12, 13, 14	syl3anc 1398	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵) ∈ 𝑆)
16		difunielsiga 34676	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵) ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) ∈ 𝑆)
17	3, 15, 16	syl2anc 596	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) ∈ 𝑆)
18	10, 17	eqeltrrd 2861	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)

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