Step | Hyp | Ref
| Expression |
1 | | inex1g 5243 |
. . 3
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V) |
2 | 1 | adantr 481 |
. 2
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V) |
3 | | inss2 4163 |
. . 3
⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 |
4 | 3 | a1i 11 |
. 2
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) |
5 | | simpr 485 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆) |
6 | | pwidg 4555 |
. . . . 5
⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴) |
7 | 5, 6 | syl 17 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴) |
8 | 5, 7 | elind 4128 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴)) |
9 | | simpll 764 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran
sigAlgebra) |
10 | | simplr 766 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆) |
11 | | inss1 4162 |
. . . . . . 7
⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆 |
12 | | simpr 485 |
. . . . . . 7
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) |
13 | 11, 12 | sselid 3919 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆) |
14 | | difelsiga 32101 |
. . . . . 6
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆) |
15 | 9, 10, 13, 14 | syl3anc 1370 |
. . . . 5
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆) |
16 | | difss 4066 |
. . . . . . 7
⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 |
17 | | elpwg 4536 |
. . . . . . 7
⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴)) |
18 | 16, 17 | mpbiri 257 |
. . . . . 6
⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴) |
19 | 15, 18 | syl 17 |
. . . . 5
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴) |
20 | 15, 19 | elind 4128 |
. . . 4
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴)) |
21 | 20 | ralrimiva 3103 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴)) |
22 | | simplll 772 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran
sigAlgebra) |
23 | | simplr 766 |
. . . . . . . 8
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) |
24 | | elpwi 4542 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴)) |
25 | | sstr 3929 |
. . . . . . . . . 10
⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥 ⊆ 𝑆) |
26 | 11, 25 | mpan2 688 |
. . . . . . . . 9
⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝑆) |
27 | 23, 24, 26 | 3syl 18 |
. . . . . . . 8
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝑆) |
28 | | elpwg 4536 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)) |
29 | 28 | biimpar 478 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥 ⊆ 𝑆) → 𝑥 ∈ 𝒫 𝑆) |
30 | 23, 27, 29 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆) |
31 | | simpr 485 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω) |
32 | | sigaclcu 32085 |
. . . . . . 7
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝑆) |
33 | 22, 30, 31, 32 | syl3anc 1370 |
. . . . . 6
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝑆) |
34 | | sstr 3929 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴) |
35 | 3, 34 | mpan2 688 |
. . . . . . . 8
⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴) |
36 | 23, 24, 35 | 3syl 18 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝒫 𝐴) |
37 | | sspwuni 5029 |
. . . . . . . 8
⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥
⊆ 𝐴) |
38 | | vuniex 7592 |
. . . . . . . . 9
⊢ ∪ 𝑥
∈ V |
39 | 38 | elpw 4537 |
. . . . . . . 8
⊢ (∪ 𝑥
∈ 𝒫 𝐴 ↔
∪ 𝑥 ⊆ 𝐴) |
40 | 37, 39 | bitr4i 277 |
. . . . . . 7
⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥
∈ 𝒫 𝐴) |
41 | 36, 40 | sylib 217 |
. . . . . 6
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝒫 𝐴) |
42 | 33, 41 | elind 4128 |
. . . . 5
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴)) |
43 | 42 | ex 413 |
. . . 4
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴))) |
44 | 43 | ralrimiva 3103 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴))) |
45 | 8, 21, 44 | 3jca 1127 |
. 2
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴)))) |
46 | | issiga 32080 |
. . 3
⊢ ((𝑆 ∩ 𝒫 𝐴) ∈ V → ((𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴) ↔ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴)))))) |
47 | 46 | biimpar 478 |
. 2
⊢ (((𝑆 ∩ 𝒫 𝐴) ∈ V ∧ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴))))) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴)) |
48 | 2, 4, 45, 47 | syl12anc 834 |
1
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴)) |