| Step | Hyp | Ref
| Expression |
| 1 | | inex1g 5319 |
. . 3
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V) |
| 2 | 1 | adantr 480 |
. 2
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V) |
| 3 | | inss2 4238 |
. . 3
⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 |
| 4 | 3 | a1i 11 |
. 2
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) |
| 5 | | simpr 484 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆) |
| 6 | | pwidg 4620 |
. . . . 5
⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴) |
| 7 | 5, 6 | syl 17 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴) |
| 8 | 5, 7 | elind 4200 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴)) |
| 9 | | simpll 767 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran
sigAlgebra) |
| 10 | | simplr 769 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆) |
| 11 | | inss1 4237 |
. . . . . . 7
⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆 |
| 12 | | simpr 484 |
. . . . . . 7
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) |
| 13 | 11, 12 | sselid 3981 |
. . . . . 6
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆) |
| 14 | | difelsiga 34134 |
. . . . . 6
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆) |
| 15 | 9, 10, 13, 14 | syl3anc 1373 |
. . . . 5
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆) |
| 16 | | difss 4136 |
. . . . . . 7
⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 |
| 17 | | elpwg 4603 |
. . . . . . 7
⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴)) |
| 18 | 16, 17 | mpbiri 258 |
. . . . . 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 4200 |
. . . 4
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴)) |
| 21 | 20 | ralrimiva 3146 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴)) |
| 22 | | simplll 775 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran
sigAlgebra) |
| 23 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) |
| 24 | | elpwi 4607 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴)) |
| 25 | | sstr 3992 |
. . . . . . . . . 10
⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥 ⊆ 𝑆) |
| 26 | 11, 25 | mpan2 691 |
. . . . . . . . 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 4603 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)) |
| 29 | 28 | biimpar 477 |
. . . . . . . 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 484 |
. . . . . . 7
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω) |
| 32 | | sigaclcu 34118 |
. . . . . . 7
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝑆) |
| 33 | 22, 30, 31, 32 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝑆) |
| 34 | | sstr 3992 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴) |
| 35 | 3, 34 | mpan2 691 |
. . . . . . . 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 5100 |
. . . . . . . 8
⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥
⊆ 𝐴) |
| 38 | | vuniex 7759 |
. . . . . . . . 9
⊢ ∪ 𝑥
∈ V |
| 39 | 38 | elpw 4604 |
. . . . . . . 8
⊢ (∪ 𝑥
∈ 𝒫 𝐴 ↔
∪ 𝑥 ⊆ 𝐴) |
| 40 | 37, 39 | bitr4i 278 |
. . . . . . 7
⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥
∈ 𝒫 𝐴) |
| 41 | 36, 40 | sylib 218 |
. . . . . 6
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ 𝒫 𝐴) |
| 42 | 33, 41 | elind 4200 |
. . . . 5
⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴)) |
| 43 | 42 | ex 412 |
. . . 4
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴))) |
| 44 | 43 | ralrimiva 3146 |
. . 3
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥
∈ (𝑆 ∩ 𝒫
𝐴))) |
| 45 | 8, 21, 44 | 3jca 1129 |
. 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 34113 |
. . 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 477 |
. 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 837 |
1
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴)) |