Theorem difelsiga 34646

Description: A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34644 and unelsiga 34645 rather than countable intersection, and so does not use ax-ac 10462. (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 34636	. . . . . 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 34644	. . . . 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 34645	. . . 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 34644	. . 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 34619

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 7737  ax-inf2 9621

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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-oi 9483  df-dju 9907  df-card 9945  df-siga 34620

This theorem is used by:  inelsiga  34647  sigainb  34648  sigaldsys  34671  cldssbrsiga  34699  measxun2  34722  measssd  34727  measunl  34728  measiuns  34729  measiun  34730  meascnbl  34731  imambfm  34774  dya2iocbrsiga  34787  dya2icobrsiga  34788  sxbrsigalem2  34798  probdif  34932