Theorem difelsiga

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

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

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → 𝐴 ∈ 𝑆 )
2		elssuni	⊢ ( 𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆 )
3		difin2	⊢ ( 𝐴 ⊆ ∪ 𝑆 → ( 𝐴 ∖ 𝐵 ) = ( ( ∪ 𝑆 ∖ 𝐵 ) ∩ 𝐴 ) )
4	1 2 3	3syl	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ 𝐵 ) = ( ( ∪ 𝑆 ∖ 𝐵 ) ∩ 𝐴 ) )
5		isrnsigau	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) )
6	5	simprd	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) )
7	6	simp2d	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 )
8		difeq2	⊢ ( 𝑥 = 𝐵 → ( ∪ 𝑆 ∖ 𝑥 ) = ( ∪ 𝑆 ∖ 𝐵 ) )
9	8	eleq1d	⊢ ( 𝑥 = 𝐵 → ( ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ↔ ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 ) )
10	9	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 )
11	7 10	sylan	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 )
12	11	3adant2	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 )
13		intprg	⊢ ( ( ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ∩ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } = ( ( ∪ 𝑆 ∖ 𝐵 ) ∩ 𝐴 ) )
14	12 1 13	syl2anc	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ∩ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } = ( ( ∪ 𝑆 ∖ 𝐵 ) ∩ 𝐴 ) )
15	4 14	eqtr4d	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ 𝐵 ) = ∩ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } )
16		simp1	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → 𝑆 ∈ ∪ ran sigAlgebra )
17		prssi	⊢ ( ( ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ⊆ 𝑆 )
18	12 1 17	syl2anc	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ⊆ 𝑆 )
19		prex	⊢ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ∈ V
20	19	elpw	⊢ ( { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ∈ 𝒫 𝑆 ↔ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ⊆ 𝑆 )
21	18 20	sylibr	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ∈ 𝒫 𝑆 )
22		prct	⊢ ( ( ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ≼ ω )
23	12 1 22	syl2anc	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ≼ ω )
24		prnzg	⊢ ( ( ∪ 𝑆 ∖ 𝐵 ) ∈ 𝑆 → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ≠ ∅ )
25	12 24	syl	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ≠ ∅ )
26		sigaclci	⊢ ( ( ( 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	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ∩ { ( ∪ 𝑆 ∖ 𝐵 ) , 𝐴 } ∈ 𝑆 )
28	15 27	eqeltrd	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 )