Step |
Hyp |
Ref |
Expression |
1 |
|
19.28v |
⊢ ( ∀ 𝑓 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
2 |
|
r19.42v |
⊢ ( ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
3 |
|
19.26 |
⊢ ( ∀ 𝑣 ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑣 ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) |
4 |
|
19.26 |
⊢ ( ∀ 𝑣 ( ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ∧ ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) ↔ ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ∧ ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) ) |
5 |
|
jcab |
⊢ ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ↔ ( ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ∧ ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) ) |
6 |
5
|
albii |
⊢ ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ↔ ∀ 𝑣 ( ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ∧ ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) ) |
7 |
|
pwss |
⊢ ( 𝒫 𝑧 ⊆ 𝑈 ↔ ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ) |
8 |
|
pwss |
⊢ ( 𝒫 𝑧 ⊆ 𝑤 ↔ ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) |
9 |
7 8
|
anbi12i |
⊢ ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤 ) ↔ ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈 ) ∧ ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤 ) ) ) |
10 |
4 6 9
|
3bitr4i |
⊢ ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤 ) ) |
11 |
|
ralcom4 |
⊢ ( ∀ 𝑖 ∈ 𝑧 ∀ 𝑣 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ∀ 𝑣 ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
12 |
|
19.23v |
⊢ ( ∀ 𝑣 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
13 |
|
3anass |
⊢ ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ( 𝑣 ∈ 𝑈 ∧ ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ) ) |
14 |
13
|
exbii |
⊢ ( ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ) ) |
15 |
|
df-rex |
⊢ ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ) ) |
16 |
14 15
|
bitr4i |
⊢ ( ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ) |
17 |
16
|
imbi1i |
⊢ ( ( ∃ 𝑣 ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
18 |
12 17
|
bitri |
⊢ ( ∀ 𝑣 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
19 |
18
|
ralbii |
⊢ ( ∀ 𝑖 ∈ 𝑧 ∀ 𝑣 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
20 |
11 19
|
bitr3i |
⊢ ( ∀ 𝑣 ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
21 |
10 20
|
anbi12i |
⊢ ( ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑣 ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤 ) ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) |
22 |
|
anass |
⊢ ( ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤 ) ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
23 |
21 22
|
bitri |
⊢ ( ( ∀ 𝑣 ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑣 ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
24 |
3 23
|
bitri |
⊢ ( ∀ 𝑣 ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
25 |
|
dfss2 |
⊢ ( 𝑣 ⊆ 𝑧 ↔ ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) ) |
26 |
|
df-an |
⊢ ( ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ↔ ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) |
27 |
25 26
|
imbi12i |
⊢ ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ↔ ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) ) |
28 |
|
3impexp |
⊢ ( ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) |
29 |
|
biid |
⊢ ( 𝑖 ∈ 𝑢 ↔ 𝑖 ∈ 𝑢 ) |
30 |
|
expanduniss |
⊢ ( ∪ 𝑢 ⊆ 𝑤 ↔ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) |
31 |
29 30
|
expandan |
⊢ ( ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ¬ ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) |
32 |
31
|
expandrexn |
⊢ ( ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) |
33 |
32
|
imbi2i |
⊢ ( ( 𝑣 ∈ 𝑓 → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) |
34 |
33
|
imbi2i |
⊢ ( ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) |
35 |
34
|
imbi2i |
⊢ ( ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) |
36 |
28 35
|
bitri |
⊢ ( ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) |
37 |
36
|
expandral |
⊢ ( ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) |
38 |
27 37
|
expandan |
⊢ ( ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) |
39 |
38
|
albii |
⊢ ( ∀ 𝑣 ( ( 𝑣 ⊆ 𝑧 → ( 𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤 ) ) ∧ ∀ 𝑖 ∈ 𝑧 ( ( 𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) |
40 |
24 39
|
bitr3i |
⊢ ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) |
41 |
40
|
expandrex |
⊢ ( ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑈 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) |
42 |
2 41
|
bitr3i |
⊢ ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑈 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) |
43 |
42
|
albii |
⊢ ( ∀ 𝑓 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑈 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) |
44 |
1 43
|
bitr3i |
⊢ ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑈 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) |
45 |
44
|
expandral |
⊢ ( ∀ 𝑧 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑈 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑈 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑈 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) |