Step |
Hyp |
Ref |
Expression |
1 |
|
rankpwi |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝒫 𝐴 ) = suc ( rank ‘ 𝐴 ) ) |
2 |
1
|
eleq1d |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( ( rank ‘ 𝒫 𝐴 ) ∈ suc 𝐵 ↔ suc ( rank ‘ 𝐴 ) ∈ suc 𝐵 ) ) |
3 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
4 |
|
ordsucelsuc |
⊢ ( Ord 𝐵 → ( ( rank ‘ 𝐴 ) ∈ 𝐵 ↔ suc ( rank ‘ 𝐴 ) ∈ suc 𝐵 ) ) |
5 |
3 4
|
syl |
⊢ ( 𝐵 ∈ On → ( ( rank ‘ 𝐴 ) ∈ 𝐵 ↔ suc ( rank ‘ 𝐴 ) ∈ suc 𝐵 ) ) |
6 |
5
|
bicomd |
⊢ ( 𝐵 ∈ On → ( suc ( rank ‘ 𝐴 ) ∈ suc 𝐵 ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) ) |
7 |
2 6
|
sylan9bb |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ On ) → ( ( rank ‘ 𝒫 𝐴 ) ∈ suc 𝐵 ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) ) |
8 |
|
pwwf |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ↔ 𝒫 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) |
9 |
8
|
biimpi |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝒫 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) |
10 |
|
suceloni |
⊢ ( 𝐵 ∈ On → suc 𝐵 ∈ On ) |
11 |
|
r1fnon |
⊢ 𝑅1 Fn On |
12 |
|
fndm |
⊢ ( 𝑅1 Fn On → dom 𝑅1 = On ) |
13 |
11 12
|
ax-mp |
⊢ dom 𝑅1 = On |
14 |
10 13
|
eleqtrrdi |
⊢ ( 𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1 ) |
15 |
|
rankr1ag |
⊢ ( ( 𝒫 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ suc 𝐵 ∈ dom 𝑅1 ) → ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ↔ ( rank ‘ 𝒫 𝐴 ) ∈ suc 𝐵 ) ) |
16 |
9 14 15
|
syl2an |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ On ) → ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ↔ ( rank ‘ 𝒫 𝐴 ) ∈ suc 𝐵 ) ) |
17 |
13
|
eleq2i |
⊢ ( 𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On ) |
18 |
|
rankr1ag |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) ) |
19 |
17 18
|
sylan2br |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) ) |
20 |
7 16 19
|
3bitr4rd |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) |
21 |
20
|
ex |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐵 ∈ On → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) ) |
22 |
|
r1elwf |
⊢ ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) → 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) |
23 |
|
r1elwf |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → 𝒫 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) |
24 |
|
r1elssi |
⊢ ( 𝒫 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝒫 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ) |
25 |
23 24
|
syl |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → 𝒫 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ) |
26 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
27 |
|
pwexr |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → 𝐴 ∈ V ) |
28 |
|
elpwg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴 ) ) |
29 |
27 28
|
syl |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → ( 𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴 ) ) |
30 |
26 29
|
mpbiri |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → 𝐴 ∈ 𝒫 𝐴 ) |
31 |
25 30
|
sseldd |
⊢ ( 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) → 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) |
32 |
22 31
|
pm5.21ni |
⊢ ( ¬ 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) |
33 |
32
|
a1d |
⊢ ( ¬ 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐵 ∈ On → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) ) |
34 |
21 33
|
pm2.61i |
⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝒫 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) |