Step |
Hyp |
Ref |
Expression |
1 |
|
df-lim |
⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) |
2 |
|
3anass |
⊢ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) ) |
3 |
|
df-ne |
⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) |
4 |
3
|
a1i |
⊢ ( Ord 𝐴 → ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) ) |
5 |
|
orduninsuc |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ↔ ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
6 |
4 5
|
anbi12d |
⊢ ( Ord 𝐴 → ( ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ↔ ( ¬ 𝐴 = ∅ ∧ ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
7 |
|
ioran |
⊢ ( ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ↔ ( ¬ 𝐴 = ∅ ∧ ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
8 |
6 7
|
bitr4di |
⊢ ( Ord 𝐴 → ( ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ↔ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
9 |
8
|
pm5.32i |
⊢ ( ( Ord 𝐴 ∧ ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) ↔ ( Ord 𝐴 ∧ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
10 |
1 2 9
|
3bitri |
⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |