| Step |
Hyp |
Ref |
Expression |
| 1 |
|
orduninsuc |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ↔ ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) ) |
| 2 |
|
ralnex |
⊢ ( ∀ 𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) |
| 3 |
|
suceloni |
⊢ ( 𝑥 ∈ On → suc 𝑥 ∈ On ) |
| 4 |
|
eloni |
⊢ ( suc 𝑥 ∈ On → Ord suc 𝑥 ) |
| 5 |
3 4
|
syl |
⊢ ( 𝑥 ∈ On → Ord suc 𝑥 ) |
| 6 |
|
ordtri3 |
⊢ ( ( Ord 𝐴 ∧ Ord suc 𝑥 ) → ( 𝐴 = suc 𝑥 ↔ ¬ ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ) ) |
| 7 |
5 6
|
sylan2 |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝐴 = suc 𝑥 ↔ ¬ ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ) ) |
| 8 |
7
|
con2bid |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ↔ ¬ 𝐴 = suc 𝑥 ) ) |
| 9 |
|
onnbtwn |
⊢ ( 𝑥 ∈ On → ¬ ( 𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥 ) ) |
| 10 |
|
imnan |
⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥 ) ↔ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥 ) ) |
| 11 |
9 10
|
sylibr |
⊢ ( 𝑥 ∈ On → ( 𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥 ) ) |
| 12 |
11
|
con2d |
⊢ ( 𝑥 ∈ On → ( 𝐴 ∈ suc 𝑥 → ¬ 𝑥 ∈ 𝐴 ) ) |
| 13 |
|
pm2.21 |
⊢ ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) |
| 14 |
12 13
|
syl6 |
⊢ ( 𝑥 ∈ On → ( 𝐴 ∈ suc 𝑥 → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 15 |
14
|
adantl |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝐴 ∈ suc 𝑥 → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 16 |
|
ax-1 |
⊢ ( suc 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) |
| 17 |
16
|
a1i |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( suc 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 18 |
15 17
|
jaod |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 19 |
|
eloni |
⊢ ( 𝑥 ∈ On → Ord 𝑥 ) |
| 20 |
|
ordtri2or |
⊢ ( ( Ord 𝑥 ∧ Ord 𝐴 ) → ( 𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥 ) ) |
| 21 |
19 20
|
sylan |
⊢ ( ( 𝑥 ∈ On ∧ Ord 𝐴 ) → ( 𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥 ) ) |
| 22 |
21
|
ancoms |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥 ) ) |
| 23 |
22
|
orcomd |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴 ) ) |
| 24 |
23
|
adantr |
⊢ ( ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) ∧ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) → ( 𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴 ) ) |
| 25 |
|
ordsssuc2 |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝐴 ⊆ 𝑥 ↔ 𝐴 ∈ suc 𝑥 ) ) |
| 26 |
25
|
biimpd |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( 𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥 ) ) |
| 27 |
26
|
adantr |
⊢ ( ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) ∧ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) → ( 𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥 ) ) |
| 28 |
|
simpr |
⊢ ( ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) ∧ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) |
| 29 |
27 28
|
orim12d |
⊢ ( ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) ∧ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) → ( ( 𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴 ) → ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ) ) |
| 30 |
24 29
|
mpd |
⊢ ( ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) ∧ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) → ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ) |
| 31 |
30
|
ex |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) → ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ) ) |
| 32 |
18 31
|
impbid |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 33 |
8 32
|
bitr3d |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ On ) → ( ¬ 𝐴 = suc 𝑥 ↔ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 34 |
33
|
pm5.74da |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On → ¬ 𝐴 = suc 𝑥 ) ↔ ( 𝑥 ∈ On → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) ) |
| 35 |
|
impexp |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑥 ∈ 𝐴 ) → suc 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ On → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 36 |
|
simpr |
⊢ ( ( 𝑥 ∈ On ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
| 37 |
|
ordelon |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On ) |
| 38 |
37
|
ex |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ On ) ) |
| 39 |
38
|
ancrd |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ On ∧ 𝑥 ∈ 𝐴 ) ) ) |
| 40 |
36 39
|
impbid2 |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥 ∈ 𝐴 ) ↔ 𝑥 ∈ 𝐴 ) ) |
| 41 |
40
|
imbi1d |
⊢ ( Ord 𝐴 → ( ( ( 𝑥 ∈ On ∧ 𝑥 ∈ 𝐴 ) → suc 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 42 |
35 41
|
bitr3id |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On → ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ↔ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 43 |
34 42
|
bitrd |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On → ¬ 𝐴 = suc 𝑥 ) ↔ ( 𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴 ) ) ) |
| 44 |
43
|
ralbidv2 |
⊢ ( Ord 𝐴 → ( ∀ 𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) |
| 45 |
2 44
|
bitr3id |
⊢ ( Ord 𝐴 → ( ¬ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) |
| 46 |
1 45
|
bitrd |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) |