Step |
Hyp |
Ref |
Expression |
1 |
|
nlim0 |
⊢ ¬ Lim ∅ |
2 |
|
limeq |
⊢ ( 𝐴 = ∅ → ( Lim 𝐴 ↔ Lim ∅ ) ) |
3 |
1 2
|
mtbiri |
⊢ ( 𝐴 = ∅ → ¬ Lim 𝐴 ) |
4 |
3
|
necon2ai |
⊢ ( Lim 𝐴 → 𝐴 ≠ ∅ ) |
5 |
|
nlim1 |
⊢ ¬ Lim 1o |
6 |
|
limeq |
⊢ ( 𝐴 = 1o → ( Lim 𝐴 ↔ Lim 1o ) ) |
7 |
5 6
|
mtbiri |
⊢ ( 𝐴 = 1o → ¬ Lim 𝐴 ) |
8 |
7
|
necon2ai |
⊢ ( Lim 𝐴 → 𝐴 ≠ 1o ) |
9 |
|
nlim2 |
⊢ ¬ Lim 2o |
10 |
|
limeq |
⊢ ( 𝐴 = 2o → ( Lim 𝐴 ↔ Lim 2o ) ) |
11 |
9 10
|
mtbiri |
⊢ ( 𝐴 = 2o → ¬ Lim 𝐴 ) |
12 |
11
|
necon2ai |
⊢ ( Lim 𝐴 → 𝐴 ≠ 2o ) |
13 |
|
limord |
⊢ ( Lim 𝐴 → Ord 𝐴 ) |
14 |
|
ord2eln012 |
⊢ ( Ord 𝐴 → ( 2o ∈ 𝐴 ↔ ( 𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o ) ) ) |
15 |
13 14
|
syl |
⊢ ( Lim 𝐴 → ( 2o ∈ 𝐴 ↔ ( 𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o ) ) ) |
16 |
4 8 12 15
|
mpbir3and |
⊢ ( Lim 𝐴 → 2o ∈ 𝐴 ) |