| Step |
Hyp |
Ref |
Expression |
| 1 |
|
alephordilem1 |
⊢ ( 𝐴 ∈ On → ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 2 |
|
alephon |
⊢ ( ℵ ‘ suc 𝐴 ) ∈ On |
| 3 |
|
cff1 |
⊢ ( ( ℵ ‘ suc 𝐴 ) ∈ On → ∃ 𝑓 ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ) |
| 4 |
2 3
|
ax-mp |
⊢ ∃ 𝑓 ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) |
| 5 |
|
fvex |
⊢ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ V |
| 6 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑦 ) ∈ V |
| 7 |
6
|
sucex |
⊢ suc ( 𝑓 ‘ 𝑦 ) ∈ V |
| 8 |
5 7
|
iunex |
⊢ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ∈ V |
| 9 |
|
f1f |
⊢ ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) → 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) |
| 10 |
9
|
ad2antrr |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) |
| 11 |
|
simplr |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) |
| 12 |
2
|
oneli |
⊢ ( 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) → 𝑥 ∈ On ) |
| 13 |
|
ffvelcdm |
⊢ ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ) → ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ) |
| 14 |
|
onelon |
⊢ ( ( ( ℵ ‘ suc 𝐴 ) ∈ On ∧ ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( 𝑓 ‘ 𝑦 ) ∈ On ) |
| 15 |
2 13 14
|
sylancr |
⊢ ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ) → ( 𝑓 ‘ 𝑦 ) ∈ On ) |
| 16 |
|
onsssuc |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑓 ‘ 𝑦 ) ∈ On ) → ( 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 17 |
15 16
|
sylan2 |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ) ) → ( 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 18 |
17
|
anassrs |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) ∧ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ) → ( 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 19 |
18
|
rexbidva |
⊢ ( ( 𝑥 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) → ( ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ∈ suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 20 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ∈ suc ( 𝑓 ‘ 𝑦 ) ) |
| 21 |
19 20
|
bitr4di |
⊢ ( ( 𝑥 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) → ( ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 22 |
21
|
ancoms |
⊢ ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑥 ∈ On ) → ( ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 23 |
12 22
|
sylan2 |
⊢ ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 24 |
23
|
ralbidva |
⊢ ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) → ( ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 25 |
|
dfss3 |
⊢ ( ( ℵ ‘ suc 𝐴 ) ⊆ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ↔ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) 𝑥 ∈ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) |
| 26 |
24 25
|
bitr4di |
⊢ ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) → ( ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ( ℵ ‘ suc 𝐴 ) ⊆ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 27 |
26
|
biimpa |
⊢ ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) → ( ℵ ‘ suc 𝐴 ) ⊆ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) |
| 28 |
10 11 27
|
syl2anc |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → ( ℵ ‘ suc 𝐴 ) ⊆ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) |
| 29 |
|
ssdomg |
⊢ ( ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ∈ V → ( ( ℵ ‘ suc 𝐴 ) ⊆ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) → ( ℵ ‘ suc 𝐴 ) ≼ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) ) |
| 30 |
8 28 29
|
mpsyl |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ) |
| 31 |
|
simprl |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → 𝐴 ∈ On ) |
| 32 |
|
onsuc |
⊢ ( 𝐴 ∈ On → suc 𝐴 ∈ On ) |
| 33 |
|
alephislim |
⊢ ( suc 𝐴 ∈ On ↔ Lim ( ℵ ‘ suc 𝐴 ) ) |
| 34 |
|
limsuc |
⊢ ( Lim ( ℵ ‘ suc 𝐴 ) → ( ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ↔ suc ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
| 35 |
33 34
|
sylbi |
⊢ ( suc 𝐴 ∈ On → ( ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ↔ suc ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
| 36 |
32 35
|
syl |
⊢ ( 𝐴 ∈ On → ( ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ↔ suc ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
| 37 |
|
breq1 |
⊢ ( 𝑧 = suc ( 𝑓 ‘ 𝑦 ) → ( 𝑧 ≺ ( ℵ ‘ suc 𝐴 ) ↔ suc ( 𝑓 ‘ 𝑦 ) ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 38 |
|
alephcard |
⊢ ( card ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) |
| 39 |
|
iscard |
⊢ ( ( card ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ↔ ( ( ℵ ‘ suc 𝐴 ) ∈ On ∧ ∀ 𝑧 ∈ ( ℵ ‘ suc 𝐴 ) 𝑧 ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 40 |
39
|
simprbi |
⊢ ( ( card ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) → ∀ 𝑧 ∈ ( ℵ ‘ suc 𝐴 ) 𝑧 ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 41 |
38 40
|
ax-mp |
⊢ ∀ 𝑧 ∈ ( ℵ ‘ suc 𝐴 ) 𝑧 ≺ ( ℵ ‘ suc 𝐴 ) |
| 42 |
37 41
|
vtoclri |
⊢ ( suc ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) → suc ( 𝑓 ‘ 𝑦 ) ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 43 |
|
alephsucdom |
⊢ ( 𝐴 ∈ On → ( suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ↔ suc ( 𝑓 ‘ 𝑦 ) ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 44 |
42 43
|
imbitrrid |
⊢ ( 𝐴 ∈ On → ( suc ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) → suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 45 |
36 44
|
sylbid |
⊢ ( 𝐴 ∈ On → ( ( 𝑓 ‘ 𝑦 ) ∈ ( ℵ ‘ suc 𝐴 ) → suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 46 |
13 45
|
syl5 |
⊢ ( 𝐴 ∈ On → ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ∧ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ) → suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 47 |
46
|
expdimp |
⊢ ( ( 𝐴 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) → ( 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) → suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 48 |
47
|
ralrimiv |
⊢ ( ( 𝐴 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) → ∀ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) |
| 49 |
|
iundom |
⊢ ( ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ V ∧ ∀ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ℵ ‘ 𝐴 ) ) → ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 50 |
5 48 49
|
sylancr |
⊢ ( ( 𝐴 ∈ On ∧ 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⟶ ( ℵ ‘ suc 𝐴 ) ) → ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 51 |
31 10 50
|
syl2anc |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 52 |
|
domtr |
⊢ ( ( ( ℵ ‘ suc 𝐴 ) ≼ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ∧ ∪ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) suc ( 𝑓 ‘ 𝑦 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 53 |
30 51 52
|
syl2anc |
⊢ ( ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 54 |
53
|
expcom |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) ) |
| 55 |
54
|
exlimdv |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ∃ 𝑓 ( 𝑓 : ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) –1-1→ ( ℵ ‘ suc 𝐴 ) ∧ ∀ 𝑥 ∈ ( ℵ ‘ suc 𝐴 ) ∃ 𝑦 ∈ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) 𝑥 ⊆ ( 𝑓 ‘ 𝑦 ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) ) |
| 56 |
4 55
|
mpi |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ) |
| 57 |
|
alephgeom |
⊢ ( 𝐴 ∈ On ↔ ω ⊆ ( ℵ ‘ 𝐴 ) ) |
| 58 |
|
alephon |
⊢ ( ℵ ‘ 𝐴 ) ∈ On |
| 59 |
|
infxpen |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ On ∧ ω ⊆ ( ℵ ‘ 𝐴 ) ) → ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ≈ ( ℵ ‘ 𝐴 ) ) |
| 60 |
58 59
|
mpan |
⊢ ( ω ⊆ ( ℵ ‘ 𝐴 ) → ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ≈ ( ℵ ‘ 𝐴 ) ) |
| 61 |
57 60
|
sylbi |
⊢ ( 𝐴 ∈ On → ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ≈ ( ℵ ‘ 𝐴 ) ) |
| 62 |
|
breq1 |
⊢ ( 𝑧 = ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) → ( 𝑧 ≺ ( ℵ ‘ suc 𝐴 ) ↔ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 63 |
62 41
|
vtoclri |
⊢ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 64 |
|
alephsucdom |
⊢ ( 𝐴 ∈ On → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ↔ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 65 |
63 64
|
imbitrrid |
⊢ ( 𝐴 ∈ On → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 66 |
|
fvex |
⊢ ( ℵ ‘ 𝐴 ) ∈ V |
| 67 |
66
|
xpdom1 |
⊢ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ) |
| 68 |
65 67
|
syl6 |
⊢ ( 𝐴 ∈ On → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ) ) |
| 69 |
|
domentr |
⊢ ( ( ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ∧ ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ≈ ( ℵ ‘ 𝐴 ) ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) |
| 70 |
69
|
expcom |
⊢ ( ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) ≈ ( ℵ ‘ 𝐴 ) → ( ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ( ℵ ‘ 𝐴 ) × ( ℵ ‘ 𝐴 ) ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 71 |
61 68 70
|
sylsyld |
⊢ ( 𝐴 ∈ On → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) ) |
| 72 |
71
|
imp |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) |
| 73 |
|
domtr |
⊢ ( ( ( ℵ ‘ suc 𝐴 ) ≼ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ∧ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) × ( ℵ ‘ 𝐴 ) ) ≼ ( ℵ ‘ 𝐴 ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ℵ ‘ 𝐴 ) ) |
| 74 |
56 72 73
|
syl2anc |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ( ℵ ‘ suc 𝐴 ) ≼ ( ℵ ‘ 𝐴 ) ) |
| 75 |
|
domnsym |
⊢ ( ( ℵ ‘ suc 𝐴 ) ≼ ( ℵ ‘ 𝐴 ) → ¬ ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 76 |
74 75
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) → ¬ ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ suc 𝐴 ) ) |
| 77 |
76
|
ex |
⊢ ( 𝐴 ∈ On → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ¬ ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ suc 𝐴 ) ) ) |
| 78 |
1 77
|
mt2d |
⊢ ( 𝐴 ∈ On → ¬ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ) |
| 79 |
|
cfon |
⊢ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ On |
| 80 |
|
cfle |
⊢ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⊆ ( ℵ ‘ suc 𝐴 ) |
| 81 |
|
onsseleq |
⊢ ( ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ On ∧ ( ℵ ‘ suc 𝐴 ) ∈ On ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ⊆ ( ℵ ‘ suc 𝐴 ) ↔ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ∨ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) ) ) |
| 82 |
80 81
|
mpbii |
⊢ ( ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ On ∧ ( ℵ ‘ suc 𝐴 ) ∈ On ) → ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ∨ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) ) |
| 83 |
79 2 82
|
mp2an |
⊢ ( ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) ∨ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) |
| 84 |
83
|
ori |
⊢ ( ¬ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) ∈ ( ℵ ‘ suc 𝐴 ) → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) |
| 85 |
78 84
|
syl |
⊢ ( 𝐴 ∈ On → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) |
| 86 |
|
cf0 |
⊢ ( cf ‘ ∅ ) = ∅ |
| 87 |
|
alephfnon |
⊢ ℵ Fn On |
| 88 |
87
|
fndmi |
⊢ dom ℵ = On |
| 89 |
88
|
eleq2i |
⊢ ( suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On ) |
| 90 |
|
onsucb |
⊢ ( 𝐴 ∈ On ↔ suc 𝐴 ∈ On ) |
| 91 |
89 90
|
bitr4i |
⊢ ( suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On ) |
| 92 |
|
ndmfv |
⊢ ( ¬ suc 𝐴 ∈ dom ℵ → ( ℵ ‘ suc 𝐴 ) = ∅ ) |
| 93 |
91 92
|
sylnbir |
⊢ ( ¬ 𝐴 ∈ On → ( ℵ ‘ suc 𝐴 ) = ∅ ) |
| 94 |
93
|
fveq2d |
⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( cf ‘ ∅ ) ) |
| 95 |
86 94 93
|
3eqtr4a |
⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) ) |
| 96 |
85 95
|
pm2.61i |
⊢ ( cf ‘ ( ℵ ‘ suc 𝐴 ) ) = ( ℵ ‘ suc 𝐴 ) |