Step |
Hyp |
Ref |
Expression |
1 |
|
eltg4i |
⊢ ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → 𝑥 = ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) |
2 |
|
inex1g |
⊢ ( 𝐵 ∈ On → ( 𝐵 ∩ 𝒫 𝑥 ) ∈ V ) |
3 |
|
onss |
⊢ ( 𝐵 ∈ On → 𝐵 ⊆ On ) |
4 |
|
ssinss1 |
⊢ ( 𝐵 ⊆ On → ( 𝐵 ∩ 𝒫 𝑥 ) ⊆ On ) |
5 |
3 4
|
syl |
⊢ ( 𝐵 ∈ On → ( 𝐵 ∩ 𝒫 𝑥 ) ⊆ On ) |
6 |
|
ssonuni |
⊢ ( ( 𝐵 ∩ 𝒫 𝑥 ) ∈ V → ( ( 𝐵 ∩ 𝒫 𝑥 ) ⊆ On → ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ∈ On ) ) |
7 |
2 5 6
|
sylc |
⊢ ( 𝐵 ∈ On → ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ∈ On ) |
8 |
|
eleq1 |
⊢ ( 𝑥 = ∪ ( 𝐵 ∩ 𝒫 𝑥 ) → ( 𝑥 ∈ On ↔ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ∈ On ) ) |
9 |
8
|
biimprd |
⊢ ( 𝑥 = ∪ ( 𝐵 ∩ 𝒫 𝑥 ) → ( ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ∈ On → 𝑥 ∈ On ) ) |
10 |
1 7 9
|
syl2imc |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → 𝑥 ∈ On ) ) |
11 |
|
onuni |
⊢ ( 𝐵 ∈ On → ∪ 𝐵 ∈ On ) |
12 |
|
suceloni |
⊢ ( ∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On ) |
13 |
11 12
|
syl |
⊢ ( 𝐵 ∈ On → suc ∪ 𝐵 ∈ On ) |
14 |
10 13
|
jctird |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → ( 𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On ) ) ) |
15 |
|
tg1 |
⊢ ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → 𝑥 ⊆ ∪ 𝐵 ) |
16 |
15
|
a1i |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → 𝑥 ⊆ ∪ 𝐵 ) ) |
17 |
|
sucidg |
⊢ ( ∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵 ) |
18 |
11 17
|
syl |
⊢ ( 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵 ) |
19 |
16 18
|
jctird |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → ( 𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵 ) ) ) |
20 |
|
ontr2 |
⊢ ( ( 𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On ) → ( ( 𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵 ) → 𝑥 ∈ suc ∪ 𝐵 ) ) |
21 |
14 19 20
|
syl6c |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) → 𝑥 ∈ suc ∪ 𝐵 ) ) |
22 |
|
elsuci |
⊢ ( 𝑥 ∈ suc ∪ 𝐵 → ( 𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵 ) ) |
23 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
24 |
|
orduniss |
⊢ ( Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵 ) |
25 |
23 24
|
syl |
⊢ ( 𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵 ) |
26 |
|
bastg |
⊢ ( 𝐵 ∈ On → 𝐵 ⊆ ( topGen ‘ 𝐵 ) ) |
27 |
25 26
|
sstrd |
⊢ ( 𝐵 ∈ On → ∪ 𝐵 ⊆ ( topGen ‘ 𝐵 ) ) |
28 |
27
|
sseld |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
29 |
|
ssid |
⊢ 𝐵 ⊆ 𝐵 |
30 |
|
eltg3i |
⊢ ( ( 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵 ) → ∪ 𝐵 ∈ ( topGen ‘ 𝐵 ) ) |
31 |
29 30
|
mpan2 |
⊢ ( 𝐵 ∈ On → ∪ 𝐵 ∈ ( topGen ‘ 𝐵 ) ) |
32 |
|
eleq1a |
⊢ ( ∪ 𝐵 ∈ ( topGen ‘ 𝐵 ) → ( 𝑥 = ∪ 𝐵 → 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
33 |
31 32
|
syl |
⊢ ( 𝐵 ∈ On → ( 𝑥 = ∪ 𝐵 → 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
34 |
28 33
|
jaod |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵 ) → 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
35 |
22 34
|
syl5 |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ suc ∪ 𝐵 → 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
36 |
21 35
|
impbid |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ 𝑥 ∈ suc ∪ 𝐵 ) ) |
37 |
36
|
eqrdv |
⊢ ( 𝐵 ∈ On → ( topGen ‘ 𝐵 ) = suc ∪ 𝐵 ) |