Step |
Hyp |
Ref |
Expression |
1 |
|
alephordilem1 |
|- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) ) |
2 |
|
alephon |
|- ( aleph ` suc A ) e. On |
3 |
|
cff1 |
|- ( ( aleph ` suc A ) e. On -> E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) ) |
4 |
2 3
|
ax-mp |
|- E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) |
5 |
|
fvex |
|- ( cf ` ( aleph ` suc A ) ) e. _V |
6 |
|
fvex |
|- ( f ` y ) e. _V |
7 |
6
|
sucex |
|- suc ( f ` y ) e. _V |
8 |
5 7
|
iunex |
|- U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) e. _V |
9 |
|
f1f |
|- ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) -> f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) |
10 |
9
|
ad2antrr |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) |
11 |
|
simplr |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) |
12 |
2
|
oneli |
|- ( x e. ( aleph ` suc A ) -> x e. On ) |
13 |
|
ffvelrn |
|- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( f ` y ) e. ( aleph ` suc A ) ) |
14 |
|
onelon |
|- ( ( ( aleph ` suc A ) e. On /\ ( f ` y ) e. ( aleph ` suc A ) ) -> ( f ` y ) e. On ) |
15 |
2 13 14
|
sylancr |
|- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( f ` y ) e. On ) |
16 |
|
onsssuc |
|- ( ( x e. On /\ ( f ` y ) e. On ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) ) |
17 |
15 16
|
sylan2 |
|- ( ( x e. On /\ ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) ) |
18 |
17
|
anassrs |
|- ( ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) ) |
19 |
18
|
rexbidva |
|- ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> E. y e. ( cf ` ( aleph ` suc A ) ) x e. suc ( f ` y ) ) ) |
20 |
|
eliun |
|- ( x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) <-> E. y e. ( cf ` ( aleph ` suc A ) ) x e. suc ( f ` y ) ) |
21 |
19 20
|
bitr4di |
|- ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
22 |
21
|
ancoms |
|- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ x e. On ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
23 |
12 22
|
sylan2 |
|- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ x e. ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
24 |
23
|
ralbidva |
|- ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) -> ( A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> A. x e. ( aleph ` suc A ) x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
25 |
|
dfss3 |
|- ( ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) <-> A. x e. ( aleph ` suc A ) x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) |
26 |
24 25
|
bitr4di |
|- ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) -> ( A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
27 |
26
|
biimpa |
|- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) |
28 |
10 11 27
|
syl2anc |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) |
29 |
|
ssdomg |
|- ( U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) e. _V -> ( ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) -> ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) ) |
30 |
8 28 29
|
mpsyl |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) |
31 |
|
simprl |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> A e. On ) |
32 |
|
suceloni |
|- ( A e. On -> suc A e. On ) |
33 |
|
alephislim |
|- ( suc A e. On <-> Lim ( aleph ` suc A ) ) |
34 |
|
limsuc |
|- ( Lim ( aleph ` suc A ) -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) ) |
35 |
33 34
|
sylbi |
|- ( suc A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) ) |
36 |
32 35
|
syl |
|- ( A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) ) |
37 |
|
breq1 |
|- ( z = suc ( f ` y ) -> ( z ~< ( aleph ` suc A ) <-> suc ( f ` y ) ~< ( aleph ` suc A ) ) ) |
38 |
|
alephcard |
|- ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) |
39 |
|
iscard |
|- ( ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) <-> ( ( aleph ` suc A ) e. On /\ A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A ) ) ) |
40 |
39
|
simprbi |
|- ( ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) -> A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A ) ) |
41 |
38 40
|
ax-mp |
|- A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A ) |
42 |
37 41
|
vtoclri |
|- ( suc ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~< ( aleph ` suc A ) ) |
43 |
|
alephsucdom |
|- ( A e. On -> ( suc ( f ` y ) ~<_ ( aleph ` A ) <-> suc ( f ` y ) ~< ( aleph ` suc A ) ) ) |
44 |
42 43
|
syl5ibr |
|- ( A e. On -> ( suc ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) ) |
45 |
36 44
|
sylbid |
|- ( A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) ) |
46 |
13 45
|
syl5 |
|- ( A e. On -> ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) ) |
47 |
46
|
expdimp |
|- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( y e. ( cf ` ( aleph ` suc A ) ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) ) |
48 |
47
|
ralrimiv |
|- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> A. y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( aleph ` A ) ) |
49 |
|
iundom |
|- ( ( ( cf ` ( aleph ` suc A ) ) e. _V /\ A. y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( aleph ` A ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
50 |
5 48 49
|
sylancr |
|- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
51 |
31 10 50
|
syl2anc |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
52 |
|
domtr |
|- ( ( ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) /\ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
53 |
30 51 52
|
syl2anc |
|- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
54 |
53
|
expcom |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) ) |
55 |
54
|
exlimdv |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) ) |
56 |
4 55
|
mpi |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) |
57 |
|
alephgeom |
|- ( A e. On <-> _om C_ ( aleph ` A ) ) |
58 |
|
alephon |
|- ( aleph ` A ) e. On |
59 |
|
infxpen |
|- ( ( ( aleph ` A ) e. On /\ _om C_ ( aleph ` A ) ) -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) ) |
60 |
58 59
|
mpan |
|- ( _om C_ ( aleph ` A ) -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) ) |
61 |
57 60
|
sylbi |
|- ( A e. On -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) ) |
62 |
|
breq1 |
|- ( z = ( cf ` ( aleph ` suc A ) ) -> ( z ~< ( aleph ` suc A ) <-> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) ) ) |
63 |
62 41
|
vtoclri |
|- ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) ) |
64 |
|
alephsucdom |
|- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) <-> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) ) ) |
65 |
63 64
|
syl5ibr |
|- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) ) ) |
66 |
|
fvex |
|- ( aleph ` A ) e. _V |
67 |
66
|
xpdom1 |
|- ( ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) ) |
68 |
65 67
|
syl6 |
|- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) ) ) |
69 |
|
domentr |
|- ( ( ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) /\ ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) |
70 |
69
|
expcom |
|- ( ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) -> ( ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) ) |
71 |
61 68 70
|
sylsyld |
|- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) ) |
72 |
71
|
imp |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) |
73 |
|
domtr |
|- ( ( ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) /\ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) -> ( aleph ` suc A ) ~<_ ( aleph ` A ) ) |
74 |
56 72 73
|
syl2anc |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( aleph ` suc A ) ~<_ ( aleph ` A ) ) |
75 |
|
domnsym |
|- ( ( aleph ` suc A ) ~<_ ( aleph ` A ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) ) |
76 |
74 75
|
syl |
|- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) ) |
77 |
76
|
ex |
|- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) ) ) |
78 |
1 77
|
mt2d |
|- ( A e. On -> -. ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) |
79 |
|
cfon |
|- ( cf ` ( aleph ` suc A ) ) e. On |
80 |
|
cfle |
|- ( cf ` ( aleph ` suc A ) ) C_ ( aleph ` suc A ) |
81 |
|
onsseleq |
|- ( ( ( cf ` ( aleph ` suc A ) ) e. On /\ ( aleph ` suc A ) e. On ) -> ( ( cf ` ( aleph ` suc A ) ) C_ ( aleph ` suc A ) <-> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) ) ) |
82 |
80 81
|
mpbii |
|- ( ( ( cf ` ( aleph ` suc A ) ) e. On /\ ( aleph ` suc A ) e. On ) -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) ) |
83 |
79 2 82
|
mp2an |
|- ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) |
84 |
83
|
ori |
|- ( -. ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) |
85 |
78 84
|
syl |
|- ( A e. On -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) |
86 |
|
cf0 |
|- ( cf ` (/) ) = (/) |
87 |
|
alephfnon |
|- aleph Fn On |
88 |
87
|
fndmi |
|- dom aleph = On |
89 |
88
|
eleq2i |
|- ( suc A e. dom aleph <-> suc A e. On ) |
90 |
|
sucelon |
|- ( A e. On <-> suc A e. On ) |
91 |
89 90
|
bitr4i |
|- ( suc A e. dom aleph <-> A e. On ) |
92 |
|
ndmfv |
|- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) ) |
93 |
91 92
|
sylnbir |
|- ( -. A e. On -> ( aleph ` suc A ) = (/) ) |
94 |
93
|
fveq2d |
|- ( -. A e. On -> ( cf ` ( aleph ` suc A ) ) = ( cf ` (/) ) ) |
95 |
86 94 93
|
3eqtr4a |
|- ( -. A e. On -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) |
96 |
85 95
|
pm2.61i |
|- ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) |