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Theorem alephreg 8413
Description: A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
Assertion
Ref Expression
alephreg  |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )

Proof of Theorem alephreg
Dummy variables  f  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephordilem1 7910 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
2 alephon 7906 . . . . . . . . 9  |-  ( aleph ` 
suc  A )  e.  On
3 cff1 8094 . . . . . . . . 9  |-  ( (
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 ) ) )
42, 3ax-mp 8 . . . . . . . 8  |-  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 5701 . . . . . . . . . . . . 13  |-  ( cf `  ( aleph `  suc  A ) )  e.  _V
6 fvex 5701 . . . . . . . . . . . . . 14  |-  ( f `
 y )  e. 
_V
76sucex 4750 . . . . . . . . . . . . 13  |-  suc  (
f `  y )  e.  _V
85, 7iunex 5950 . . . . . . . . . . . 12  |-  U_ y  e.  ( cf `  ( aleph `  suc  A ) ) suc  ( f `
 y )  e. 
_V
9 f1f 5598 . . . . . . . . . . . . . 14  |-  ( f : ( cf `  ( aleph `  suc  A ) ) -1-1-> ( aleph `  suc  A )  ->  f :
( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )
109ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( 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 732 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) )
122oneli 4648 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( aleph `  suc  A )  ->  x  e.  On )
13 ffvelrn 5827 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  ( aleph `  suc  A ) )
14 onelon 4566 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( aleph `  suc  A )  e.  On  /\  (
f `  y )  e.  ( aleph `  suc  A ) )  ->  ( f `  y )  e.  On )
152, 13, 14sylancr 645 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  On )
16 onsssuc 4628 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  ( f `  y
)  e.  On )  ->  ( x  C_  ( f `  y
)  <->  x  e.  suc  ( f `  y
) ) )
1715, 16sylan2 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 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
) ) )
1817anassrs 630 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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
) ) )
1918rexbidva 2683 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 4057 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  U_ y  e.  ( cf `  ( aleph `  suc  A ) ) suc  ( f `
 y )  <->  E. y  e.  ( cf `  ( aleph `  suc  A ) ) x  e.  suc  ( f `  y
) )
2119, 20syl6bbr 255 . . . . . . . . . . . . . . . . . 18  |-  ( ( 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 ) ) )
2221ancoms 440 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 ) ) )
2312, 22sylan2 461 . . . . . . . . . . . . . . . 16  |-  ( ( 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 ) ) )
2423ralbidva 2682 . . . . . . . . . . . . . . 15  |-  ( 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 3298 . . . . . . . . . . . . . . 15  |-  ( (
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 ) )
2624, 25syl6bbr 255 . . . . . . . . . . . . . 14  |-  ( 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 ) ) )
2726biimpa 471 . . . . . . . . . . . . 13  |-  ( ( 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 ) )
2810, 11, 27syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( 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 7112 . . . . . . . . . . . 12  |-  ( 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 ) ) )
308, 28, 29mpsyl 61 . . . . . . . . . . 11  |-  ( ( ( 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 733 . . . . . . . . . . . 12  |-  ( ( ( 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 4752 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  suc  A  e.  On )
33 alephislim 7920 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
A  e.  On  <->  Lim  ( aleph ` 
suc  A ) )
34 limsuc 4788 . . . . . . . . . . . . . . . . . . 19  |-  ( Lim  ( aleph `  suc  A )  ->  ( ( f `
 y )  e.  ( aleph `  suc  A )  <->  suc  ( f `  y
)  e.  ( aleph ` 
suc  A ) ) )
3533, 34sylbi 188 . . . . . . . . . . . . . . . . . 18  |-  ( suc 
A  e.  On  ->  ( ( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
3632, 35syl 16 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
37 breq1 4175 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  suc  ( f `
 y )  -> 
( z  ~<  ( aleph `  suc  A )  <->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) ) )
38 alephcard 7907 . . . . . . . . . . . . . . . . . . . 20  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
39 iscard 7818 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. z  e.  ( aleph ` 
suc  A ) z 
~<  ( aleph `  suc  A ) ) )
4039simprbi 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  ->  A. z  e.  (
aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A ) )
4138, 40ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  A. z  e.  ( aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A )
4237, 41vtoclri 2986 . . . . . . . . . . . . . . . . . 18  |-  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) )
43 alephsucdom 7916 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  ~<_  ( aleph `  A
)  <->  suc  ( f `  y )  ~<  ( aleph `  suc  A ) ) )
4442, 43syl5ibr 213 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4536, 44sylbid 207 . . . . . . . . . . . . . . . 16  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4613, 45syl5 30 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  (
( f : ( cf `  ( aleph ` 
suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph ` 
suc  A ) ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4746expdimp 427 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )  ->  (
y  e.  ( cf `  ( aleph `  suc  A ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4847ralrimiv 2748 . . . . . . . . . . . . 13  |-  ( ( 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 8373 . . . . . . . . . . . . 13  |-  ( ( ( 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
) ) )
505, 48, 49sylancr 645 . . . . . . . . . . . 12  |-  ( ( 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 ) ) )
5131, 10, 50syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( 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 7119 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
5330, 51, 52syl2anc 643 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
5453expcom 425 . . . . . . . . 9  |-  ( ( 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 ) ) ) )
5554exlimdv 1643 . . . . . . . 8  |-  ( ( 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 ) ) ) )
564, 55mpi 17 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) ) )
57 alephgeom 7919 . . . . . . . . . 10  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
58 alephon 7906 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  On
59 infxpen 7852 . . . . . . . . . . 11  |-  ( ( ( aleph `  A )  e.  On  /\  om  C_  ( aleph `  A ) )  ->  ( ( aleph `  A )  X.  ( aleph `  A ) ) 
~~  ( aleph `  A
) )
6058, 59mpan 652 . . . . . . . . . 10  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  X.  ( aleph `  A ) )  ~~  ( aleph `  A )
)
6157, 60sylbi 188 . . . . . . . . 9  |-  ( A  e.  On  ->  (
( aleph `  A )  X.  ( aleph `  A )
)  ~~  ( aleph `  A ) )
62 breq1 4175 . . . . . . . . . . . 12  |-  ( z  =  ( cf `  ( aleph `  suc  A ) )  ->  ( z  ~<  ( aleph `  suc  A )  <-> 
( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6362, 41vtoclri 2986 . . . . . . . . . . 11  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  ->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) )
64 alephsucdom 7916 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
)  <->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6563, 64syl5ibr 213 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
) ) )
66 fvex 5701 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  _V
6766xpdom1 7166 . . . . . . . . . 10  |-  ( ( cf `  ( aleph ` 
suc  A ) )  ~<_  ( aleph `  A )  ->  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  ( ( aleph `  A )  X.  ( aleph `  A )
) )
6865, 67syl6 31 . . . . . . . . 9  |-  ( 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 7125 . . . . . . . . . 10  |-  ( ( ( ( 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 ) )
7069expcom 425 . . . . . . . . 9  |-  ( ( ( 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 )
) )
7161, 68, 70sylsyld 54 . . . . . . . 8  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) ) )
7271imp 419 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  (
( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )
73 domtr 7119 . . . . . . 7  |-  ( ( ( aleph `  suc  A )  ~<_  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  /\  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A
) )
7456, 72, 73syl2anc 643 . . . . . 6  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A )
)
75 domnsym 7192 . . . . . 6  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  A )  ->  -.  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )
7674, 75syl 16 . . . . 5  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
7776ex 424 . . . 4  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) ) )
781, 77mt2d 111 . . 3  |-  ( A  e.  On  ->  -.  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )
79 cfon 8091 . . . . 5  |-  ( cf `  ( aleph `  suc  A ) )  e.  On
80 cfle 8090 . . . . . 6  |-  ( cf `  ( aleph `  suc  A ) )  C_  ( aleph ` 
suc  A )
81 onsseleq 4582 . . . . . 6  |-  ( ( ( 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 ) ) ) )
8280, 81mpbii 203 . . . . 5  |-  ( ( ( 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 ) ) )
8379, 2, 82mp2an 654 . . . 4  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  \/  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8483ori 365 . . 3  |-  ( -.  ( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8578, 84syl 16 . 2  |-  ( A  e.  On  ->  ( cf `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A ) )
86 cf0 8087 . . 3  |-  ( cf `  (/) )  =  (/)
87 alephfnon 7902 . . . . . . . 8  |-  aleph  Fn  On
88 fndm 5503 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
8987, 88ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
9089eleq2i 2468 . . . . . 6  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
91 sucelon 4756 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
9290, 91bitr4i 244 . . . . 5  |-  ( suc 
A  e.  dom  aleph  <->  A  e.  On )
93 ndmfv 5714 . . . . 5  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
9492, 93sylnbir 299 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  suc  A )  =  (/) )
9594fveq2d 5691 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( cf `  (/) ) )
9686, 95, 943eqtr4a 2462 . 2  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( aleph `  suc  A ) )
9785, 96pm2.61i 158 1  |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   (/)c0 3588   U_ciun 4053   class class class wbr 4172   Oncon0 4541   Lim wlim 4542   suc csuc 4543   omcom 4804    X. cxp 4835   dom cdm 4837    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778   alephcale 7779   cfccf 7780
This theorem is referenced by:  pwcfsdom  8414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783  df-cf 7784  df-acn 7785  df-ac 7953
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