Step |
Hyp |
Ref |
Expression |
1 |
|
eluni2 |
⊢ ( 𝑥 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ) |
2 |
|
ssel |
⊢ ( 𝐴 ⊆ On → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ On ) ) |
3 |
|
onelss |
⊢ ( 𝑦 ∈ On → ( 𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦 ) ) |
4 |
2 3
|
syl6 |
⊢ ( 𝐴 ⊆ On → ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦 ) ) ) |
5 |
|
anc2r |
⊢ ( ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦 ) ) → ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → ( 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ⊆ On → ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → ( 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) ) |
7 |
|
ssuni |
⊢ ( ( 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ⊆ ∪ 𝐴 ) |
8 |
6 7
|
syl8 |
⊢ ( 𝐴 ⊆ On → ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴 ) ) ) |
9 |
8
|
rexlimdv |
⊢ ( 𝐴 ⊆ On → ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴 ) ) |
10 |
1 9
|
syl5bi |
⊢ ( 𝐴 ⊆ On → ( 𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴 ) ) |
11 |
10
|
ralrimiv |
⊢ ( 𝐴 ⊆ On → ∀ 𝑥 ∈ ∪ 𝐴 𝑥 ⊆ ∪ 𝐴 ) |
12 |
|
dftr3 |
⊢ ( Tr ∪ 𝐴 ↔ ∀ 𝑥 ∈ ∪ 𝐴 𝑥 ⊆ ∪ 𝐴 ) |
13 |
11 12
|
sylibr |
⊢ ( 𝐴 ⊆ On → Tr ∪ 𝐴 ) |
14 |
|
onelon |
⊢ ( ( 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦 ) → 𝑥 ∈ On ) |
15 |
14
|
ex |
⊢ ( 𝑦 ∈ On → ( 𝑥 ∈ 𝑦 → 𝑥 ∈ On ) ) |
16 |
2 15
|
syl6 |
⊢ ( 𝐴 ⊆ On → ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝑦 → 𝑥 ∈ On ) ) ) |
17 |
16
|
rexlimdv |
⊢ ( 𝐴 ⊆ On → ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On ) ) |
18 |
1 17
|
syl5bi |
⊢ ( 𝐴 ⊆ On → ( 𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On ) ) |
19 |
18
|
ssrdv |
⊢ ( 𝐴 ⊆ On → ∪ 𝐴 ⊆ On ) |
20 |
|
ordon |
⊢ Ord On |
21 |
|
trssord |
⊢ ( ( Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On ) → Ord ∪ 𝐴 ) |
22 |
21
|
3exp |
⊢ ( Tr ∪ 𝐴 → ( ∪ 𝐴 ⊆ On → ( Ord On → Ord ∪ 𝐴 ) ) ) |
23 |
20 22
|
mpii |
⊢ ( Tr ∪ 𝐴 → ( ∪ 𝐴 ⊆ On → Ord ∪ 𝐴 ) ) |
24 |
13 19 23
|
sylc |
⊢ ( 𝐴 ⊆ On → Ord ∪ 𝐴 ) |