Step |
Hyp |
Ref |
Expression |
1 |
|
isrnsiga |
⊢ ( 𝑆 ∈ ∪ ran sigAlgebra ↔ ( 𝑆 ∈ V ∧ ∃ 𝑜 ( 𝑆 ⊆ 𝒫 𝑜 ∧ ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) ) ) |
2 |
1
|
simprbi |
⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∃ 𝑜 ( 𝑆 ⊆ 𝒫 𝑜 ∧ ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) ) |
3 |
|
3simpa |
⊢ ( ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) → ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑆 ⊆ 𝒫 𝑜 ∧ ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) → ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ) ) |
5 |
4
|
eximi |
⊢ ( ∃ 𝑜 ( 𝑆 ⊆ 𝒫 𝑜 ∧ ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) → ∃ 𝑜 ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ) ) |
6 |
|
difeq2 |
⊢ ( 𝑥 = 𝑜 → ( 𝑜 ∖ 𝑥 ) = ( 𝑜 ∖ 𝑜 ) ) |
7 |
|
difid |
⊢ ( 𝑜 ∖ 𝑜 ) = ∅ |
8 |
6 7
|
eqtrdi |
⊢ ( 𝑥 = 𝑜 → ( 𝑜 ∖ 𝑥 ) = ∅ ) |
9 |
8
|
eleq1d |
⊢ ( 𝑥 = 𝑜 → ( ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ↔ ∅ ∈ 𝑆 ) ) |
10 |
9
|
rspcva |
⊢ ( ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ) → ∅ ∈ 𝑆 ) |
11 |
10
|
exlimiv |
⊢ ( ∃ 𝑜 ( 𝑜 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑜 ∖ 𝑥 ) ∈ 𝑆 ) → ∅ ∈ 𝑆 ) |
12 |
2 5 11
|
3syl |
⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆 ) |