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Theorem alephreg 9007
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 8504 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
2 alephon 8500 . . . . . . . . 9  |-  ( aleph ` 
suc  A )  e.  On
3 cff1 8688 . . . . . . . . 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 5 . . . . . . . 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 5875 . . . . . . . . . . . . 13  |-  ( cf `  ( aleph `  suc  A ) )  e.  _V
6 fvex 5875 . . . . . . . . . . . . . 14  |-  ( f `
 y )  e. 
_V
76sucex 6638 . . . . . . . . . . . . 13  |-  suc  (
f `  y )  e.  _V
85, 7iunex 6773 . . . . . . . . . . . 12  |-  U_ y  e.  ( cf `  ( aleph `  suc  A ) ) suc  ( f `
 y )  e. 
_V
9 f1f 5779 . . . . . . . . . . . . . 14  |-  ( f : ( cf `  ( aleph `  suc  A ) ) -1-1-> ( aleph `  suc  A )  ->  f :
( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )
109ad2antrr 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 ) ) )  ->  f :
( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )
11 simplr 762 . . . . . . . . . . . . 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 5530 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( aleph `  suc  A )  ->  x  e.  On )
13 ffvelrn 6020 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  ( aleph `  suc  A ) )
14 onelon 5448 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( aleph `  suc  A )  e.  On  /\  (
f `  y )  e.  ( aleph `  suc  A ) )  ->  ( f `  y )  e.  On )
152, 13, 14sylancr 669 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  On )
16 onsssuc 5510 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  ( f `  y
)  e.  On )  ->  ( x  C_  ( f `  y
)  <->  x  e.  suc  ( f `  y
) ) )
1715, 16sylan2 477 . . . . . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . . . . . 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 2898 . . . . . . . . . . . . . . . . . . 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 4283 . . . . . . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 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 3422 . . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . 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 667 . . . . . . . . . . . 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 7615 . . . . . . . . . . . 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 65 . . . . . . . . . . 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 764 . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  suc  A  e.  On )
33 alephislim 8514 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
A  e.  On  <->  Lim  ( aleph ` 
suc  A ) )
34 limsuc 6676 . . . . . . . . . . . . . . . . . . 19  |-  ( Lim  ( aleph `  suc  A )  ->  ( ( f `
 y )  e.  ( aleph `  suc  A )  <->  suc  ( f `  y
)  e.  ( aleph ` 
suc  A ) ) )
3533, 34sylbi 199 . . . . . . . . . . . . . . . . . 18  |-  ( suc 
A  e.  On  ->  ( ( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
3632, 35syl 17 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  <->  suc  ( f `
 y )  e.  ( aleph `  suc  A ) ) )
37 breq1 4405 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  suc  ( f `
 y )  -> 
( z  ~<  ( aleph `  suc  A )  <->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) ) )
38 alephcard 8501 . . . . . . . . . . . . . . . . . . . 20  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
39 iscard 8409 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. z  e.  ( aleph ` 
suc  A ) z 
~<  ( aleph `  suc  A ) ) )
4039simprbi 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  ->  A. z  e.  (
aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A ) )
4138, 40ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  A. z  e.  ( aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A )
4237, 41vtoclri 3124 . . . . . . . . . . . . . . . . . 18  |-  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) )
43 alephsucdom 8510 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  ~<_  ( aleph `  A
)  <->  suc  ( f `  y )  ~<  ( aleph `  suc  A ) ) )
4442, 43syl5ibr 225 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4536, 44sylbid 219 . . . . . . . . . . . . . . . 16  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4613, 45syl5 33 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  (
( f : ( cf `  ( aleph ` 
suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph ` 
suc  A ) ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4746expdimp 439 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )  ->  (
y  e.  ( cf `  ( aleph `  suc  A ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4847ralrimiv 2800 . . . . . . . . . . . . 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 8967 . . . . . . . . . . . . 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 669 . . . . . . . . . . . 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 667 . . . . . . . . . . 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 7622 . . . . . . . . . . 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 667 . . . . . . . . . 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 437 . . . . . . . . 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 1779 . . . . . . . 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 20 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) ) )
57 alephgeom 8513 . . . . . . . . . 10  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
58 alephon 8500 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  On
59 infxpen 8445 . . . . . . . . . . 11  |-  ( ( ( aleph `  A )  e.  On  /\  om  C_  ( aleph `  A ) )  ->  ( ( aleph `  A )  X.  ( aleph `  A ) ) 
~~  ( aleph `  A
) )
6058, 59mpan 676 . . . . . . . . . 10  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  X.  ( aleph `  A ) )  ~~  ( aleph `  A )
)
6157, 60sylbi 199 . . . . . . . . 9  |-  ( A  e.  On  ->  (
( aleph `  A )  X.  ( aleph `  A )
)  ~~  ( aleph `  A ) )
62 breq1 4405 . . . . . . . . . . . 12  |-  ( z  =  ( cf `  ( aleph `  suc  A ) )  ->  ( z  ~<  ( aleph `  suc  A )  <-> 
( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6362, 41vtoclri 3124 . . . . . . . . . . 11  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  ->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) )
64 alephsucdom 8510 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
)  <->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6563, 64syl5ibr 225 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
) ) )
66 fvex 5875 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  _V
6766xpdom1 7671 . . . . . . . . . 10  |-  ( ( cf `  ( aleph ` 
suc  A ) )  ~<_  ( aleph `  A )  ->  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  ( ( aleph `  A )  X.  ( aleph `  A )
) )
6865, 67syl6 34 . . . . . . . . 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 7628 . . . . . . . . . 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 437 . . . . . . . . 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 58 . . . . . . . 8  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) ) )
7271imp 431 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  (
( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )
73 domtr 7622 . . . . . . 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 667 . . . . . 6  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A )
)
75 domnsym 7698 . . . . . 6  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  A )  ->  -.  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )
7674, 75syl 17 . . . . 5  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
7776ex 436 . . . 4  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) ) )
781, 77mt2d 121 . . 3  |-  ( A  e.  On  ->  -.  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )
79 cfon 8685 . . . . 5  |-  ( cf `  ( aleph `  suc  A ) )  e.  On
80 cfle 8684 . . . . . 6  |-  ( cf `  ( aleph `  suc  A ) )  C_  ( aleph ` 
suc  A )
81 onsseleq 5464 . . . . . 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 215 . . . . 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 678 . . . 4  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  \/  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8483ori 377 . . 3  |-  ( -.  ( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8578, 84syl 17 . 2  |-  ( A  e.  On  ->  ( cf `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A ) )
86 cf0 8681 . . 3  |-  ( cf `  (/) )  =  (/)
87 alephfnon 8496 . . . . . . . 8  |-  aleph  Fn  On
88 fndm 5675 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
8987, 88ax-mp 5 . . . . . . 7  |-  dom  aleph  =  On
9089eleq2i 2521 . . . . . 6  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
91 sucelon 6644 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
9290, 91bitr4i 256 . . . . 5  |-  ( suc 
A  e.  dom  aleph  <->  A  e.  On )
93 ndmfv 5889 . . . . 5  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
9492, 93sylnbir 309 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  suc  A )  =  (/) )
9594fveq2d 5869 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( cf `  (/) ) )
9686, 95, 943eqtr4a 2511 . 2  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( aleph `  suc  A ) )
9785, 96pm2.61i 168 1  |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045    C_ wss 3404   (/)c0 3731   U_ciun 4278   class class class wbr 4402    X. cxp 4832   dom cdm 4834   Oncon0 5423   Lim wlim 5424   suc csuc 5425    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   ` cfv 5582   omcom 6692    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568   cardccrd 8369   alephcale 8370   cfccf 8371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-har 8073  df-card 8373  df-aleph 8374  df-cf 8375  df-acn 8376  df-ac 8547
This theorem is referenced by:  pwcfsdom  9008
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