Step |
Hyp |
Ref |
Expression |
1 |
|
harcl |
⊢ ( har ‘ ( ℵ ‘ 𝐴 ) ) ∈ On |
2 |
|
alephon |
⊢ ( ℵ ‘ 𝐴 ) ∈ On |
3 |
|
onenon |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ On → ( ℵ ‘ 𝐴 ) ∈ dom card ) |
4 |
|
harsdom |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ dom card → ( ℵ ‘ 𝐴 ) ≺ ( har ‘ ( ℵ ‘ 𝐴 ) ) ) |
5 |
2 3 4
|
mp2b |
⊢ ( ℵ ‘ 𝐴 ) ≺ ( har ‘ ( ℵ ‘ 𝐴 ) ) |
6 |
|
simp1 |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → 𝐴 ∈ On ) |
7 |
|
alephgeom |
⊢ ( 𝐴 ∈ On ↔ ω ⊆ ( ℵ ‘ 𝐴 ) ) |
8 |
6 7
|
sylib |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ω ⊆ ( ℵ ‘ 𝐴 ) ) |
9 |
|
ssdomg |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ On → ( ω ⊆ ( ℵ ‘ 𝐴 ) → ω ≼ ( ℵ ‘ 𝐴 ) ) ) |
10 |
2 8 9
|
mpsyl |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ω ≼ ( ℵ ‘ 𝐴 ) ) |
11 |
|
simp2 |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( ℵ ‘ 𝐴 ) ∈ GCH ) |
12 |
|
alephsuc |
⊢ ( 𝐴 ∈ On → ( ℵ ‘ suc 𝐴 ) = ( har ‘ ( ℵ ‘ 𝐴 ) ) ) |
13 |
6 12
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( ℵ ‘ suc 𝐴 ) = ( har ‘ ( ℵ ‘ 𝐴 ) ) ) |
14 |
|
simp3 |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( ℵ ‘ suc 𝐴 ) ∈ GCH ) |
15 |
13 14
|
eqeltrrd |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( har ‘ ( ℵ ‘ 𝐴 ) ) ∈ GCH ) |
16 |
|
gchpwdom |
⊢ ( ( ω ≼ ( ℵ ‘ 𝐴 ) ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( har ‘ ( ℵ ‘ 𝐴 ) ) ∈ GCH ) → ( ( ℵ ‘ 𝐴 ) ≺ ( har ‘ ( ℵ ‘ 𝐴 ) ) ↔ 𝒫 ( ℵ ‘ 𝐴 ) ≼ ( har ‘ ( ℵ ‘ 𝐴 ) ) ) ) |
17 |
10 11 15 16
|
syl3anc |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( ( ℵ ‘ 𝐴 ) ≺ ( har ‘ ( ℵ ‘ 𝐴 ) ) ↔ 𝒫 ( ℵ ‘ 𝐴 ) ≼ ( har ‘ ( ℵ ‘ 𝐴 ) ) ) ) |
18 |
5 17
|
mpbii |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → 𝒫 ( ℵ ‘ 𝐴 ) ≼ ( har ‘ ( ℵ ‘ 𝐴 ) ) ) |
19 |
|
ondomen |
⊢ ( ( ( har ‘ ( ℵ ‘ 𝐴 ) ) ∈ On ∧ 𝒫 ( ℵ ‘ 𝐴 ) ≼ ( har ‘ ( ℵ ‘ 𝐴 ) ) ) → 𝒫 ( ℵ ‘ 𝐴 ) ∈ dom card ) |
20 |
1 18 19
|
sylancr |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → 𝒫 ( ℵ ‘ 𝐴 ) ∈ dom card ) |
21 |
|
gchaleph |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ 𝒫 ( ℵ ‘ 𝐴 ) ∈ dom card ) → ( ℵ ‘ suc 𝐴 ) ≈ 𝒫 ( ℵ ‘ 𝐴 ) ) |
22 |
20 21
|
syld3an3 |
⊢ ( ( 𝐴 ∈ On ∧ ( ℵ ‘ 𝐴 ) ∈ GCH ∧ ( ℵ ‘ suc 𝐴 ) ∈ GCH ) → ( ℵ ‘ suc 𝐴 ) ≈ 𝒫 ( ℵ ‘ 𝐴 ) ) |