Step |
Hyp |
Ref |
Expression |
1 |
|
ordtoplem.1 |
⊢ ( ∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆 ) |
2 |
|
df-ne |
⊢ ( 𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴 ) |
3 |
|
ordeleqon |
⊢ ( Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On ) ) |
4 |
|
unon |
⊢ ∪ On = On |
5 |
4
|
eqcomi |
⊢ On = ∪ On |
6 |
|
id |
⊢ ( 𝐴 = On → 𝐴 = On ) |
7 |
|
unieq |
⊢ ( 𝐴 = On → ∪ 𝐴 = ∪ On ) |
8 |
5 6 7
|
3eqtr4a |
⊢ ( 𝐴 = On → 𝐴 = ∪ 𝐴 ) |
9 |
8
|
orim2i |
⊢ ( ( 𝐴 ∈ On ∨ 𝐴 = On ) → ( 𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴 ) ) |
10 |
3 9
|
sylbi |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴 ) ) |
11 |
10
|
orcomd |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On ) ) |
12 |
11
|
ord |
⊢ ( Ord 𝐴 → ( ¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On ) ) |
13 |
|
orduniorsuc |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) ) |
14 |
13
|
ord |
⊢ ( Ord 𝐴 → ( ¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴 ) ) |
15 |
|
onuni |
⊢ ( 𝐴 ∈ On → ∪ 𝐴 ∈ On ) |
16 |
|
eleq1a |
⊢ ( suc ∪ 𝐴 ∈ 𝑆 → ( 𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆 ) ) |
17 |
15 1 16
|
3syl |
⊢ ( 𝐴 ∈ On → ( 𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆 ) ) |
18 |
12 14 17
|
syl6c |
⊢ ( Ord 𝐴 → ( ¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ 𝑆 ) ) |
19 |
2 18
|
syl5bi |
⊢ ( Ord 𝐴 → ( 𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆 ) ) |