Theorem difelsiga 34432

Description: A sigma-algebra is closed under class differences. (Contributed by Thierry Arnoux, 13-Sep-2016.)

Assertion

Ref	Expression
difelsiga	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)

Proof of Theorem difelsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1151	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
2		elssuni 4899	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
3		difin2 4255	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑆 → (𝐴 ∖ 𝐵) = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴))
4	1, 2, 3	3syl 18	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴))
5		isrnsigau 34426	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
6	5	simprd 499	. . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
7	6	simp2d 1157	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
8		difeq2 4076	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ 𝐵))
9	8	eleq1d 2849	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆))
10	9	rspccva 3582	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
11	7, 10	sylan 589	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
12	11	3adant2 1145	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
13		intprg 4941	. . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴))
14	12, 1, 13	syl2anc 593	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} = ((∪ 𝑆 ∖ 𝐵) ∩ 𝐴))
15	4, 14	eqtr4d 2802	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) = ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴})
16		simp1 1150	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
17		prssi 4781	. . . . 5 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆)
18	12, 1, 17	syl2anc 593	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆)
19		prex 5397	. . . . 5 ⊢ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ V
20	19	elpw 4561	. . . 4 ⊢ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆 ↔ {(∪ 𝑆 ∖ 𝐵), 𝐴} ⊆ 𝑆)
21	18, 20	sylibr 236	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆)
22		prct 32917	. . . 4 ⊢ (((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω)
23	12, 1, 22	syl2anc 593	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω)
24		prnzg 4739	. . . 4 ⊢ ((∪ 𝑆 ∖ 𝐵) ∈ 𝑆 → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅)
25	12, 24	syl 17	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅)
26		sigaclci 34431	. . 3 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝒫 𝑆) ∧ ({(∪ 𝑆 ∖ 𝐵), 𝐴} ≼ ω ∧ {(∪ 𝑆 ∖ 𝐵), 𝐴} ≠ ∅)) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆)
27	16, 21, 23, 25, 26	syl22anc 849	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∩ {(∪ 𝑆 ∖ 𝐵), 𝐴} ∈ 𝑆)
28	15, 27	eqeltrd 2864	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {cpr 4586  ∪ cuni 4867  ∩ cint 4907   class class class wbr 5102  ran crn 5650  ωcom 7848   ≼ cdom 8927  sigAlgebracsiga 34407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-ac2 10422

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  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-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-oi 9460  df-dju 9861  df-card 9899  df-acn 9902  df-ac 10074  df-siga 34408

This theorem is referenced by:  inelsiga  34434  sigainb  34435  sigaldsys  34458  cldssbrsiga  34486  measxun2  34509  measssd  34514  measunl  34515  measiuns  34516  measiun  34517  meascnbl  34518  imambfm  34561  dya2iocbrsiga  34574  dya2icobrsiga  34575  sxbrsigalem2  34585  probdif  34719