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Theorem alephreg 8953
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 8450 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
2 alephon 8446 . . . . . . . . 9  |-  ( aleph ` 
suc  A )  e.  On
3 cff1 8634 . . . . . . . . 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 5874 . . . . . . . . . . . . 13  |-  ( cf `  ( aleph `  suc  A ) )  e.  _V
6 fvex 5874 . . . . . . . . . . . . . 14  |-  ( f `
 y )  e. 
_V
76sucex 6624 . . . . . . . . . . . . 13  |-  suc  (
f `  y )  e.  _V
85, 7iunex 6761 . . . . . . . . . . . 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 725 . . . . . . . . . . . . 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 754 . . . . . . . . . . . . 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 4985 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( aleph `  suc  A )  ->  x  e.  On )
13 ffvelrn 6017 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  ( aleph `  suc  A ) )
14 onelon 4903 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( aleph `  suc  A )  e.  On  /\  (
f `  y )  e.  ( aleph `  suc  A ) )  ->  ( f `  y )  e.  On )
152, 13, 14sylancr 663 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  On )
16 onsssuc 4965 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  ( f `  y
)  e.  On )  ->  ( x  C_  ( f `  y
)  <->  x  e.  suc  ( f `  y
) ) )
1715, 16sylan2 474 . . . . . . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . . . . . . 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 2970 . . . . . . . . . . . . . . . . . . 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 4330 . . . . . . . . . . . . . . . . . . 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 263 . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 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 2900 . . . . . . . . . . . . . . 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 3494 . . . . . . . . . . . . . . 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 263 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 7558 . . . . . . . . . . . 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 63 . . . . . . . . . . 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 755 . . . . . . . . . . . 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 6626 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  suc  A  e.  On )
33 alephislim 8460 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
A  e.  On  <->  Lim  ( aleph ` 
suc  A ) )
34 limsuc 6662 . . . . . . . . . . . . . . . . . . 19  |-  ( Lim  ( aleph `  suc  A )  ->  ( ( f `
 y )  e.  ( aleph `  suc  A )  <->  suc  ( f `  y
)  e.  ( aleph ` 
suc  A ) ) )
3533, 34sylbi 195 . . . . . . . . . . . . . . . . . 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 4450 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  suc  ( f `
 y )  -> 
( z  ~<  ( aleph `  suc  A )  <->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) ) )
38 alephcard 8447 . . . . . . . . . . . . . . . . . . . 20  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
39 iscard 8352 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. z  e.  ( aleph ` 
suc  A ) z 
~<  ( aleph `  suc  A ) ) )
4039simprbi 464 . . . . . . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . . . . 18  |-  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) )
43 alephsucdom 8456 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  ~<_  ( aleph `  A
)  <->  suc  ( f `  y )  ~<  ( aleph `  suc  A ) ) )
4442, 43syl5ibr 221 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4536, 44sylbid 215 . . . . . . . . . . . . . . . 16  |-  ( A  e.  On  ->  (
( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4613, 45syl5 32 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  (
( f : ( cf `  ( aleph ` 
suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph ` 
suc  A ) ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4746expdimp 437 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )  ->  (
y  e.  ( cf `  ( aleph `  suc  A ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4847ralrimiv 2876 . . . . . . . . . . . . 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 8913 . . . . . . . . . . . . 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 663 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 7565 . . . . . . . . . . 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 661 . . . . . . . . . 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 435 . . . . . . . . 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 1700 . . . . . . . 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 8459 . . . . . . . . . 10  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
58 alephon 8446 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  On
59 infxpen 8388 . . . . . . . . . . 11  |-  ( ( ( aleph `  A )  e.  On  /\  om  C_  ( aleph `  A ) )  ->  ( ( aleph `  A )  X.  ( aleph `  A ) ) 
~~  ( aleph `  A
) )
6058, 59mpan 670 . . . . . . . . . 10  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  X.  ( aleph `  A ) )  ~~  ( aleph `  A )
)
6157, 60sylbi 195 . . . . . . . . 9  |-  ( A  e.  On  ->  (
( aleph `  A )  X.  ( aleph `  A )
)  ~~  ( aleph `  A ) )
62 breq1 4450 . . . . . . . . . . . 12  |-  ( z  =  ( cf `  ( aleph `  suc  A ) )  ->  ( z  ~<  ( aleph `  suc  A )  <-> 
( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6362, 41vtoclri 3188 . . . . . . . . . . 11  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  ->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) )
64 alephsucdom 8456 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
)  <->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6563, 64syl5ibr 221 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
) ) )
66 fvex 5874 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  _V
6766xpdom1 7613 . . . . . . . . . 10  |-  ( ( cf `  ( aleph ` 
suc  A ) )  ~<_  ( aleph `  A )  ->  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  ( ( aleph `  A )  X.  ( aleph `  A )
) )
6865, 67syl6 33 . . . . . . . . 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 7571 . . . . . . . . . 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 435 . . . . . . . . 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 56 . . . . . . . 8  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) ) )
7271imp 429 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  (
( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )
73 domtr 7565 . . . . . . 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 661 . . . . . 6  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A )
)
75 domnsym 7640 . . . . . 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 434 . . . 4  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  ->  -.  ( aleph `  A )  ~<  ( aleph `  suc  A ) ) )
781, 77mt2d 117 . . 3  |-  ( A  e.  On  ->  -.  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )
79 cfon 8631 . . . . 5  |-  ( cf `  ( aleph `  suc  A ) )  e.  On
80 cfle 8630 . . . . . 6  |-  ( cf `  ( aleph `  suc  A ) )  C_  ( aleph ` 
suc  A )
81 onsseleq 4919 . . . . . 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 211 . . . . 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 672 . . . 4  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  \/  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8483ori 375 . . 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 8627 . . 3  |-  ( cf `  (/) )  =  (/)
87 alephfnon 8442 . . . . . . . 8  |-  aleph  Fn  On
88 fndm 5678 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
8987, 88ax-mp 5 . . . . . . 7  |-  dom  aleph  =  On
9089eleq2i 2545 . . . . . 6  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
91 sucelon 6630 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
9290, 91bitr4i 252 . . . . 5  |-  ( suc 
A  e.  dom  aleph  <->  A  e.  On )
93 ndmfv 5888 . . . . 5  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
9492, 93sylnbir 307 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  suc  A )  =  (/) )
9594fveq2d 5868 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( cf `  (/) ) )
9686, 95, 943eqtr4a 2534 . 2  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( aleph `  suc  A ) )
9785, 96pm2.61i 164 1  |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   U_ciun 4325   class class class wbr 4447   Oncon0 4878   Lim wlim 4879   suc csuc 4880    X. cxp 4997   dom cdm 4999    Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   ` cfv 5586   omcom 6678    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512   cardccrd 8312   alephcale 8313   cfccf 8314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-ac2 8839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-har 7980  df-card 8316  df-aleph 8317  df-cf 8318  df-acn 8319  df-ac 8493
This theorem is referenced by:  pwcfsdom  8954
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