Step |
Hyp |
Ref |
Expression |
1 |
|
dflim4 |
⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) |
2 |
|
suceq |
⊢ ( 𝑥 = 𝐵 → suc 𝑥 = suc 𝐵 ) |
3 |
2
|
eleq1d |
⊢ ( 𝑥 = 𝐵 → ( suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴 ) ) |
4 |
3
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) ) |
5 |
4
|
3ad2ant3 |
⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) ) |
6 |
1 5
|
sylbi |
⊢ ( Lim 𝐴 → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) ) |
7 |
|
limord |
⊢ ( Lim 𝐴 → Ord 𝐴 ) |
8 |
|
ordtr |
⊢ ( Ord 𝐴 → Tr 𝐴 ) |
9 |
|
trsuc |
⊢ ( ( Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) |
10 |
9
|
ex |
⊢ ( Tr 𝐴 → ( suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴 ) ) |
11 |
7 8 10
|
3syl |
⊢ ( Lim 𝐴 → ( suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴 ) ) |
12 |
6 11
|
impbid |
⊢ ( Lim 𝐴 → ( 𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴 ) ) |