Theorem difelsiga 34630

Description: A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34628 and unelsiga 34629 rather than countable intersection, and so does not use ax-ac 10464. (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 4248	. . . 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 34620	. . . . . 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 4218	. . . . 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 4076	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) ∖ 𝐵) = (𝐴 ∖ 𝐵))
10	2, 9	eqtrd 2797	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) = (𝐴 ∖ 𝐵))
11		difunielsiga 34628	. . . . 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 34629	. . . 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 34628	. . 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 2863	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 3899   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4870  ran crn 5660  sigAlgebracsiga 34603

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-dju 9909  df-card 9947  df-siga 34604

This theorem is used by:  inelsiga  34631  sigainb  34632  sigaldsys  34655  cldssbrsiga  34683  measxun2  34706  measssd  34711  measunl  34712  measiuns  34713  measiun  34714  meascnbl  34715  imambfm  34758  dya2iocbrsiga  34771  dya2icobrsiga  34772  sxbrsigalem2  34782  probdif  34916