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Theorem alephreg 8137
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
StepHypRef Expression
1 alephordilem1 7633 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
2 alephon 7629 . . . . . . . . 9  |-  ( aleph ` 
suc  A )  e.  On
3 cff1 7817 . . . . . . . . 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 10 . . . . . . . 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 5437 . . . . . . . . . . . . 13  |-  ( cf `  ( aleph `  suc  A ) )  e.  _V
6 fvex 5437 . . . . . . . . . . . . . 14  |-  ( f `
 y )  e. 
_V
76sucex 4539 . . . . . . . . . . . . 13  |-  suc  (
f `  y )  e.  _V
85, 7iunex 5669 . . . . . . . . . . . 12  |-  U_ y  e.  ( cf `  ( aleph `  suc  A ) ) suc  ( f `
 y )  e. 
_V
9 f1f 5340 . . . . . . . . . . . . . 14  |-  ( f : ( cf `  ( aleph `  suc  A ) ) -1-1-> ( aleph `  suc  A )  ->  f :
( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )
109ad2antrr 709 . . . . . . . . . . . . 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 734 . . . . . . . . . . . . 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 4437 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( aleph `  suc  A )  ->  x  e.  On )
13 ffvelrn 5562 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  ( aleph `  suc  A ) )
14 onelon 4354 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( aleph `  suc  A )  e.  On  /\  (
f `  y )  e.  ( aleph `  suc  A ) )  ->  ( f `  y )  e.  On )
152, 13, 14sylancr 647 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A )  /\  y  e.  ( cf `  ( aleph `  suc  A ) ) )  ->  (
f `  y )  e.  On )
16 onsssuc 4417 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  ( f `  y
)  e.  On )  ->  ( x  C_  ( f `  y
)  <->  x  e.  suc  ( f `  y
) ) )
1715, 16sylan2 462 . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . 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 2531 . . . . . . . . . . . . . . . . . . 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 3850 . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . 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 441 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 645 . . . . . . . . . . . 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 6840 . . . . . . . . . . . 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 735 . . . . . . . . . . . 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 4541 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  suc  A  e.  On )
33 alephislim 7643 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
A  e.  On  <->  Lim  ( aleph ` 
suc  A ) )
34 limsuc 4577 . . . . . . . . . . . . . . . . . . 19  |-  ( Lim  ( aleph `  suc  A )  ->  ( ( f `
 y )  e.  ( aleph `  suc  A )  <->  suc  ( f `  y
)  e.  ( aleph ` 
suc  A ) ) )
3533, 34sylbi 189 . . . . . . . . . . . . . . . . . 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 3966 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  suc  ( f `
 y )  -> 
( z  ~<  ( aleph `  suc  A )  <->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) ) )
38 alephcard 7630 . . . . . . . . . . . . . . . . . . . 20  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
39 iscard 7541 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. z  e.  ( aleph ` 
suc  A ) z 
~<  ( aleph `  suc  A ) ) )
4039simprbi 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  ->  A. z  e.  (
aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A ) )
4138, 40ax-mp 10 . . . . . . . . . . . . . . . . . . 19  |-  A. z  e.  ( aleph `  suc  A ) z  ~<  ( aleph ` 
suc  A )
4237, 41vtoclri 2809 . . . . . . . . . . . . . . . . . 18  |-  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<  ( aleph `  suc  A ) )
43 alephsucdom 7639 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  ~<_  ( aleph `  A
)  <->  suc  ( f `  y )  ~<  ( aleph `  suc  A ) ) )
4442, 43syl5ibr 214 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  ( suc  ( f `  y
)  e.  ( aleph ` 
suc  A )  ->  suc  ( f `  y
)  ~<_  ( aleph `  A
) ) )
4536, 44sylbid 208 . . . . . . . . . . . . . . . 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 428 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  f : ( cf `  ( aleph `  suc  A ) ) --> ( aleph `  suc  A ) )  ->  (
y  e.  ( cf `  ( aleph `  suc  A ) )  ->  suc  ( f `
 y )  ~<_  (
aleph `  A ) ) )
4847ralrimiv 2596 . . . . . . . . . . . . 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 8097 . . . . . . . . . . . . 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 647 . . . . . . . . . . . 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 645 . . . . . . . . . . 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 6847 . . . . . . . . . . 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 645 . . . . . . . . . 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 426 . . . . . . . . 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 1933 . . . . . . . 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 18 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( ( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) ) )
57 alephgeom 7642 . . . . . . . . . 10  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
58 alephon 7629 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  On
59 infxpen 7575 . . . . . . . . . . 11  |-  ( ( ( aleph `  A )  e.  On  /\  om  C_  ( aleph `  A ) )  ->  ( ( aleph `  A )  X.  ( aleph `  A ) ) 
~~  ( aleph `  A
) )
6058, 59mpan 654 . . . . . . . . . 10  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  X.  ( aleph `  A ) )  ~~  ( aleph `  A )
)
6157, 60sylbi 189 . . . . . . . . 9  |-  ( A  e.  On  ->  (
( aleph `  A )  X.  ( aleph `  A )
)  ~~  ( aleph `  A ) )
62 breq1 3966 . . . . . . . . . . . 12  |-  ( z  =  ( cf `  ( aleph `  suc  A ) )  ->  ( z  ~<  ( aleph `  suc  A )  <-> 
( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6362, 41vtoclri 2809 . . . . . . . . . . 11  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  ->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) )
64 alephsucdom 7639 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
)  <->  ( cf `  ( aleph `  suc  A ) )  ~<  ( aleph ` 
suc  A ) ) )
6563, 64syl5ibr 214 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( cf `  ( aleph `  suc  A ) )  e.  ( aleph ` 
suc  A )  -> 
( cf `  ( aleph `  suc  A ) )  ~<_  ( aleph `  A
) ) )
66 fvex 5437 . . . . . . . . . . 11  |-  ( aleph `  A )  e.  _V
6766xpdom1 6894 . . . . . . . . . 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 6853 . . . . . . . . . 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 426 . . . . . . . . 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 420 . . . . . . 7  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  (
( cf `  ( aleph `  suc  A ) )  X.  ( aleph `  A ) )  ~<_  (
aleph `  A ) )
73 domtr 6847 . . . . . . 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 645 . . . . . 6  |-  ( ( A  e.  On  /\  ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A ) )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  A )
)
75 domnsym 6920 . . . . . 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 425 . . . 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 7814 . . . . 5  |-  ( cf `  ( aleph `  suc  A ) )  e.  On
80 cfle 7813 . . . . . 6  |-  ( cf `  ( aleph `  suc  A ) )  C_  ( aleph ` 
suc  A )
81 onsseleq 4370 . . . . . 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 204 . . . . 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 656 . . . 4  |-  ( ( cf `  ( aleph ` 
suc  A ) )  e.  ( aleph `  suc  A )  \/  ( cf `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A ) )
8483ori 366 . . 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 7810 . . 3  |-  ( cf `  (/) )  =  (/)
87 alephfnon 7625 . . . . . . . 8  |-  aleph  Fn  On
88 fndm 5246 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
8987, 88ax-mp 10 . . . . . . 7  |-  dom  aleph  =  On
9089eleq2i 2320 . . . . . 6  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
91 sucelon 4545 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
9290, 91bitr4i 245 . . . . 5  |-  ( suc 
A  e.  dom  aleph  <->  A  e.  On )
93 ndmfv 5451 . . . . 5  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
9492, 93sylnbir 300 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  suc  A )  =  (/) )
9594fveq2d 5427 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  ( aleph ` 
suc  A ) )  =  ( cf `  (/) ) )
9686, 95, 943eqtr4a 2314 . 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 5    <-> wb 178    \/ wo 359    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   _Vcvv 2740    C_ wss 3094   (/)c0 3397   U_ciun 3846   class class class wbr 3963   Oncon0 4329   Lim wlim 4330   suc csuc 4331   omcom 4593    X. cxp 4624   dom cdm 4626    Fn wfn 4633   -->wf 4634   -1-1->wf1 4635   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795   cardccrd 7501   alephcale 7502   cfccf 7503
This theorem is referenced by:  pwcfsdom  8138
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-ac2 8022
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506  df-cf 7507  df-acn 7508  df-ac 7676
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