Step |
Hyp |
Ref |
Expression |
1 |
|
3ancomb |
⊢ ( ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴 ) ) |
2 |
|
df-3an |
⊢ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴 ) ↔ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ∧ ¬ Lim 𝐴 ) ) |
3 |
|
df-ne |
⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) |
4 |
3
|
anbi2i |
⊢ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ( Ord 𝐴 ∧ ¬ 𝐴 = ∅ ) ) |
5 |
4
|
imbi1i |
⊢ ( ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ↔ ( ( Ord 𝐴 ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
6 |
|
pm5.6 |
⊢ ( ( ( Ord 𝐴 ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ↔ ( Ord 𝐴 → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
7 |
|
iman |
⊢ ( ( Ord 𝐴 → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ↔ ¬ ( Ord 𝐴 ∧ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
8 |
5 6 7
|
3bitrri |
⊢ ( ¬ ( Ord 𝐴 ∧ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ↔ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
9 |
|
dflim3 |
⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ¬ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
10 |
8 9
|
xchnxbir |
⊢ ( ¬ Lim 𝐴 ↔ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
11 |
10
|
anbi2i |
⊢ ( ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ∧ ¬ Lim 𝐴 ) ↔ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ∧ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
12 |
1 2 11
|
3bitri |
⊢ ( ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ∧ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) ) |
13 |
|
pm3.35 |
⊢ ( ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) ∧ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) |
14 |
12 13
|
sylbi |
⊢ ( ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) |
15 |
|
eloni |
⊢ ( 𝑥 ∈ On → Ord 𝑥 ) |
16 |
|
ordsuc |
⊢ ( Ord 𝑥 ↔ Ord suc 𝑥 ) |
17 |
15 16
|
sylib |
⊢ ( 𝑥 ∈ On → Ord suc 𝑥 ) |
18 |
|
nlimsucg |
⊢ ( 𝑥 ∈ On → ¬ Lim suc 𝑥 ) |
19 |
|
nsuceq0 |
⊢ suc 𝑥 ≠ ∅ |
20 |
19
|
a1i |
⊢ ( 𝑥 ∈ On → suc 𝑥 ≠ ∅ ) |
21 |
17 18 20
|
3jca |
⊢ ( 𝑥 ∈ On → ( Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅ ) ) |
22 |
|
ordeq |
⊢ ( 𝐴 = suc 𝑥 → ( Ord 𝐴 ↔ Ord suc 𝑥 ) ) |
23 |
|
limeq |
⊢ ( 𝐴 = suc 𝑥 → ( Lim 𝐴 ↔ Lim suc 𝑥 ) ) |
24 |
23
|
notbid |
⊢ ( 𝐴 = suc 𝑥 → ( ¬ Lim 𝐴 ↔ ¬ Lim suc 𝑥 ) ) |
25 |
|
neeq1 |
⊢ ( 𝐴 = suc 𝑥 → ( 𝐴 ≠ ∅ ↔ suc 𝑥 ≠ ∅ ) ) |
26 |
22 24 25
|
3anbi123d |
⊢ ( 𝐴 = suc 𝑥 → ( ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ( Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅ ) ) ) |
27 |
21 26
|
syl5ibrcom |
⊢ ( 𝑥 ∈ On → ( 𝐴 = suc 𝑥 → ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ) ) |
28 |
27
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 → ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ) |
29 |
14 28
|
impbii |
⊢ ( ( Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) |