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Theorem alephreg 8220
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 7716 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
2 alephon 7712 . . . . . . . . 9  |-  ( aleph ` 
suc  A )  e.  On
3 cff1 7900 . . . . . . . . 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 5555 . . . . . . . . . . . . 13  |-  ( cf `  ( aleph `  suc  A ) )  e.  _V
6 fvex 5555 . . . . . . . . . . . . . 14  |-  ( f `
 y )  e. 
_V
76sucex 4618 . . . . . . . . . . . . 13  |-  suc  (
f `  y )  e.  _V
85, 7iunex 5786 . . . . . . . . . . . 12  |-  U_ y  e.  ( cf `  ( aleph `  suc  A ) ) suc  ( f `
 y )  e. 
_V
9 f1f 5453 . . . . . . . . . . . . . 14  |-  ( f : ( cf `  ( aleph `  suc  A ) ) -1-1-> ( aleph `  suc  A )  ->  f :
( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )
109ad2antrr 706 . . . . . . . . . . . . 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 731 . . . . . . . . . . . . 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 4516 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( aleph `  suc  A )  ->  x  e.  On )
13 ffvelrn 5679 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  ( aleph `  suc  A ) )
14 onelon 4433 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( aleph `  suc  A )  e.  On  /\  (
f `  y )  e.  ( aleph `  suc  A ) )  ->  ( f `  y )  e.  On )
152, 13, 14sylancr 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  On )
16 onsssuc 4496 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  ( f `  y
)  e.  On )  ->  ( x  C_  ( f `  y
)  <->  x  e.  suc  ( f `  y
) ) )
1715, 16sylan2 460 . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . . 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 2573 . . . . . . . . . . . . . . . . . . 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 3925 . . . . . . . . . . . . . . . . . . 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 254 . . . . . . . . . . . . . . . . . 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 439 . . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . . 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 2572 . . . . . . . . . . . . . . 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 3183 . . . . . . . . . . . . . . 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 254 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 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 6923 . . . . . . . . . . . 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 59 . . . . . . . . . . 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 732 . . . . . . . . . . . 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 4620 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  suc  A  e.  On )
33 alephislim 7726 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
A  e.  On  <->  Lim  ( aleph ` 
suc  A ) )
34 limsuc 4656 . . . . . . . . . . . . . . . . . . 19  |-  ( Lim  ( aleph `  suc  A )  ->  ( ( f `
 y )  e.  ( aleph `  suc  A )  <->  suc  ( f `  y
)  e.  ( aleph ` 
suc  A ) ) )
3533, 34sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( suc 
A  e.  On  ->  ( ( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
3632, 35syl 15 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
37 breq1 4042 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  suc  ( f `
 y )  -> 
( z  ~<  ( aleph `  suc  A )  <->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) ) )
38 alephcard 7713 . . . . . . . . . . . . . . . . . . . 20  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
39 iscard 7624 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. z  e.  ( aleph ` 
suc  A ) z 
~<  ( aleph `  suc  A ) ) )
4039simprbi 450 . . . . . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . . . . 18  |-  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) )
43 alephsucdom 7722 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  ~<_  ( aleph `  A
)  <->  suc  ( f `  y )  ~<  ( aleph `  suc  A ) ) )
4442, 43syl5ibr 212 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4536, 44sylbid 206 . . . . . . . . . . . . . . . 16  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4613, 45syl5 28 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  (
( f : ( cf `  ( aleph ` 
suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph ` 
suc  A ) ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4746expdimp 426 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )  ->  (
y  e.  ( cf `  ( aleph `  suc  A ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4847ralrimiv 2638 . . . . . . . . . . . . 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 8180 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 6930 . . . . . . . . . . 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 642 . . . . . . . . . 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 424 . . . . . . . . 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 1626 . . . . . . . 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 16 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) ) )
57 alephgeom 7725 . . . . . . . . . 10  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
58 alephon 7712 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  On
59 infxpen 7658 . . . . . . . . . . 11  |-  ( ( ( aleph `  A )  e.  On  /\  om  C_  ( aleph `  A ) )  ->  ( ( aleph `  A )  X.  ( aleph `  A ) ) 
~~  ( aleph `  A
) )
6058, 59mpan 651 . . . . . . . . . 10  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  X.  ( aleph `  A ) )  ~~  ( aleph `  A )
)
6157, 60sylbi 187 . . . . . . . . 9  |-  ( A  e.  On  ->  (
( aleph `  A )  X.  ( aleph `  A )
)  ~~  ( aleph `  A ) )
62 breq1 4042 . . . . . . . . . . . 12  |-  ( z  =  ( cf `  ( aleph `  suc  A ) )  ->  ( z  ~<  ( aleph `  suc  A )  <-> 
( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6362, 41vtoclri 2871 . . . . . . . . . . 11  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  ->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) )
64 alephsucdom 7722 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
)  <->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6563, 64syl5ibr 212 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
) ) )
66 fvex 5555 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  _V
6766xpdom1 6977 . . . . . . . . . 10  |-  ( ( cf `  ( aleph ` 
suc  A ) )  ~<_  ( aleph `  A )  ->  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  ( ( aleph `  A )  X.  ( aleph `  A )
) )
6865, 67syl6 29 . . . . . . . . 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 6936 . . . . . . . . . 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 424 . . . . . . . . 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 52 . . . . . . . 8  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) ) )
7271imp 418 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  (
( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )
73 domtr 6930 . . . . . . 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 642 . . . . . 6  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A )
)
75 domnsym 7003 . . . . . 6  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  A )  ->  -.  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )
7674, 75syl 15 . . . . 5  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
7776ex 423 . . . 4  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) ) )
781, 77mt2d 109 . . 3  |-  ( A  e.  On  ->  -.  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )
79 cfon 7897 . . . . 5  |-  ( cf `  ( aleph `  suc  A ) )  e.  On
80 cfle 7896 . . . . . 6  |-  ( cf `  ( aleph `  suc  A ) )  C_  ( aleph ` 
suc  A )
81 onsseleq 4449 . . . . . 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 202 . . . . 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 653 . . . 4  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  \/  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8483ori 364 . . 3  |-  ( -.  ( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8578, 84syl 15 . 2  |-  ( A  e.  On  ->  ( cf `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A ) )
86 cf0 7893 . . 3  |-  ( cf `  (/) )  =  (/)
87 alephfnon 7708 . . . . . . . 8  |-  aleph  Fn  On
88 fndm 5359 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
8987, 88ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
9089eleq2i 2360 . . . . . 6  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
91 sucelon 4624 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
9290, 91bitr4i 243 . . . . 5  |-  ( suc 
A  e.  dom  aleph  <->  A  e.  On )
93 ndmfv 5568 . . . . 5  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
9492, 93sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  suc  A )  =  (/) )
9594fveq2d 5545 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( cf `  (/) ) )
9686, 95, 943eqtr4a 2354 . 2  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( aleph `  suc  A ) )
9785, 96pm2.61i 156 1  |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U_ciun 3921   class class class wbr 4039   Oncon0 4408   Lim wlim 4409   suc csuc 4410   omcom 4672    X. cxp 4703   dom cdm 4705    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584   alephcale 7585   cfccf 7586
This theorem is referenced by:  pwcfsdom  8221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589  df-cf 7590  df-acn 7591  df-ac 7759
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