Theorem difelsiga 34837

Description: A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34835 and unelsiga 34836 rather than countable intersection, and so does not use ax-ac 10564. (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 4247	. . . 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 34827	. . . . . 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 4217	. . . . 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 4065	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐴)) ∖ 𝐵) = (𝐴 ∖ 𝐵))
10	2, 9	eqtrd 2798	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∪ 𝑆 ∖ ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵)) = (𝐴 ∖ 𝐵))
11		difunielsiga 34835	. . . . 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 34836	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (∪ 𝑆 ∖ 𝐴) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵) ∈ 𝑆)
15	3, 12, 13, 14	syl3anc 1399	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∪ 𝑆 ∖ 𝐴) ∪ 𝐵) ∈ 𝑆)
16		difunielsiga 34835	. . 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 2865	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∪ cuni 4870  ran crn 5656  sigAlgebracsiga 34810

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-dju 10009  df-card 10047  df-siga 34811

This theorem is used by:  inelsiga  34838  sigainb  34839  sigaldsys  34862  cldssbrsiga  34890  measxun2  34913  measssd  34918  measunl  34919  measiuns  34920  measiun  34921  meascnbl  34922  imambfm  34964  dya2iocbrsiga  34977  dya2icobrsiga  34978  sxbrsigalem2  34988  probdif  35122