Theorem difelsiga 34809

Description: A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34807 and unelsiga 34808 rather than countable intersection, and so does not use ax-ac 10537. (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 4244	. . . 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 34799	. . . . . 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 4214	. . . . 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 4062	. . 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 34807	. . . . 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 34808	. . . 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 34807	. . 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 1569   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  ∪ cuni 4866  ran crn 5651  sigAlgebracsiga 34782

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-oi 9504  df-dju 9982  df-card 10020  df-siga 34783

This theorem is used by:  inelsiga  34810  sigainb  34811  sigaldsys  34834  cldssbrsiga  34862  measxun2  34885  measssd  34890  measunl  34891  measiuns  34892  measiun  34893  meascnbl  34894  imambfm  34936  dya2iocbrsiga  34949  dya2icobrsiga  34950  sxbrsigalem2  34960  probdif  35094