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Theorem cardaleph 8251
Description: Given any transfinite cardinal number  A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
cardaleph  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) )
Distinct variable group:    x, A

Proof of Theorem cardaleph
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardon 8106 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2497 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 alephle 8250 . . . . . . . . 9  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
5 fveq2 5684 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
65sseq2d 3377 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  A ) ) )
76rspcev 3066 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  C_  ( aleph `  A
) )  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
84, 7mpdan 668 . . . . . . . 8  |-  ( A  e.  On  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
9 nfcv 2573 . . . . . . . . . 10  |-  F/_ x A
10 nfcv 2573 . . . . . . . . . . 11  |-  F/_ x aleph
11 nfrab1 2895 . . . . . . . . . . . 12  |-  F/_ x { x  e.  On  |  A  C_  ( aleph `  x ) }
1211nfint 4131 . . . . . . . . . . 11  |-  F/_ x |^| { x  e.  On  |  A  C_  ( aleph `  x ) }
1310, 12nffv 5691 . . . . . . . . . 10  |-  F/_ x
( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
149, 13nfss 3342 . . . . . . . . 9  |-  F/ x  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
15 fveq2 5684 . . . . . . . . . 10  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1615sseq2d 3377 . . . . . . . . 9  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( A  C_  ( aleph `  x )  <->  A 
C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
1714, 16onminsb 6405 . . . . . . . 8  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  ->  A  C_  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
183, 8, 173syl 20 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1918a1i 11 . . . . . 6  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( card `  A
)  =  A  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
20 fveq2 5684 . . . . . . . . 9  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (
aleph `  (/) ) )
21 aleph0 8228 . . . . . . . . 9  |-  ( aleph `  (/) )  =  om
2220, 21syl6eq 2485 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  om )
2322sseq1d 3376 . . . . . . 7  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  C_  A 
<->  om  C_  A )
)
2423biimprd 223 . . . . . 6  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( om  C_  A  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  C_  A
) )
2519, 24anim12d 563 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( ( card `  A )  =  A  /\  om  C_  A
)  ->  ( A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  /\  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  C_  A
) ) )
26 eqss 3364 . . . . 5  |-  ( A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  <->  ( A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  /\  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  C_  A
) )
2725, 26syl6ibr 227 . . . 4  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( ( card `  A )  =  A  /\  om  C_  A
)  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2827com12 31 . . 3  |-  ( ( ( card `  A
)  =  A  /\  om  C_  A )  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2928ancoms 453 . 2  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
30 vex 2969 . . . . . . . . . . . 12  |-  y  e. 
_V
3130sucid 4790 . . . . . . . . . . 11  |-  y  e. 
suc  y
32 eleq2 2498 . . . . . . . . . . 11  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( y  e. 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  <->  y  e.  suc  y ) )
3331, 32mpbiri 233 . . . . . . . . . 10  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
34 fveq2 5684 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
3534sseq2d 3377 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  y ) ) )
3635onnminsb 6410 . . . . . . . . . 10  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  C_  ( aleph `  y ) ) )
3733, 36syl5 32 . . . . . . . . 9  |-  ( y  e.  On  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  ->  -.  A  C_  ( aleph `  y )
) )
3837imp 429 . . . . . . . 8  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  -.  A  C_  ( aleph `  y
) )
3938adantl 466 . . . . . . 7  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  ->  -.  A  C_  ( aleph `  y ) )
40 fveq2 5684 . . . . . . . . . . 11  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  =  ( aleph `  suc  y ) )
41 alephsuc 8230 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
4240, 41sylan9eqr 2491 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (har
`  ( aleph `  y
) ) )
4342eleq2d 2504 . . . . . . . . 9  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  <-> 
A  e.  (har `  ( aleph `  y )
) ) )
4443biimpd 207 . . . . . . . 8  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  e.  (har
`  ( aleph `  y
) ) ) )
45 elharval 7770 . . . . . . . . . 10  |-  ( A  e.  (har `  ( aleph `  y ) )  <-> 
( A  e.  On  /\  A  ~<_  ( aleph `  y
) ) )
4645simprbi 464 . . . . . . . . 9  |-  ( A  e.  (har `  ( aleph `  y ) )  ->  A  ~<_  ( aleph `  y ) )
47 onenon 8111 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  A  e.  dom  card )
483, 47syl 16 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  A  e. 
dom  card )
49 alephon 8231 . . . . . . . . . . . 12  |-  ( aleph `  y )  e.  On
50 onenon 8111 . . . . . . . . . . . 12  |-  ( (
aleph `  y )  e.  On  ->  ( aleph `  y )  e.  dom  card )
5149, 50ax-mp 5 . . . . . . . . . . 11  |-  ( aleph `  y )  e.  dom  card
52 carddom2 8139 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\  ( aleph `  y )  e.  dom  card )  ->  (
( card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
5348, 51, 52sylancl 662 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
54 sseq1 3370 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( card `  ( aleph `  y )
) ) )
55 alephcard 8232 . . . . . . . . . . . 12  |-  ( card `  ( aleph `  y )
)  =  ( aleph `  y )
5655sseq2i 3374 . . . . . . . . . . 11  |-  ( A 
C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) )
5754, 56syl6bb 261 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) ) )
5853, 57bitr3d 255 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( A  ~<_  ( aleph `  y )  <->  A 
C_  ( aleph `  y
) ) )
5946, 58syl5ib 219 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  ( A  e.  (har `  ( aleph `  y ) )  ->  A  C_  ( aleph `  y ) ) )
6044, 59sylan9r 658 . . . . . . 7  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  -> 
( A  e.  (
aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  C_  ( aleph `  y )
) )
6139, 60mtod 177 . . . . . 6  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  ->  -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
6261rexlimdvaa 2836 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( E. y  e.  On  |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
63 onintrab2 6408 . . . . . . . . . . . . . 14  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  <->  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
648, 63sylib 196 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
65 onelon 4736 . . . . . . . . . . . . 13  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  -> 
y  e.  On )
6664, 65sylan 471 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  y  e.  On )
6736adantld 467 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  (
( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) ) )
6866, 67mpcom 36 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) )
6949onelssi 4819 . . . . . . . . . . 11  |-  ( A  e.  ( aleph `  y
)  ->  A  C_  ( aleph `  y ) )
7068, 69nsyl 121 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph `  y ) )
7170nrexdv 2813 . . . . . . . . 9  |-  ( A  e.  On  ->  -.  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
7271adantr 465 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
73 alephlim 8229 . . . . . . . . . . 11  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  U_ y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ( aleph `  y
) )
7464, 73sylan 471 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  U_ y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ( aleph `  y
) )
7574eleq2d 2504 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  <-> 
A  e.  U_ y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ( aleph `  y
) ) )
76 eliun 4168 . . . . . . . . 9  |-  ( A  e.  U_ y  e. 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ( aleph `  y )  <->  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
7775, 76syl6bb 261 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  <->  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) ) )
7872, 77mtbird 301 . . . . . . 7  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
7978ex 434 . . . . . 6  |-  ( A  e.  On  ->  ( Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
803, 79syl 16 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8162, 80jaod 380 . . . 4  |-  ( (
card `  A )  =  A  ->  ( ( E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
828, 17syl 16 . . . . . 6  |-  ( A  e.  On  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
83 alephon 8231 . . . . . . 7  |-  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On
84 onsseleq 4752 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On )  ->  ( A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  <->  ( A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  \/  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) ) )
8583, 84mpan2 671 . . . . . 6  |-  ( A  e.  On  ->  ( A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  <->  ( A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  \/  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) ) )
8682, 85mpbid 210 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  \/  A  =  (
aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8786ord 377 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
883, 81, 87sylsyld 56 . . 3  |-  ( (
card `  A )  =  A  ->  ( ( E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  =  (
aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8988adantl 466 . 2  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  (
( E. y  e.  On  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/ 
Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
90 eloni 4721 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
91 ordzsl 6451 . . . . . 6  |-  ( Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  <->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  \/ 
E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } ) )
92 3orass 968 . . . . . 6  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
\/  E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } )  <-> 
( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/)  \/  ( E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
9391, 92bitri 249 . . . . 5  |-  ( Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  <->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  \/  ( E. y  e.  On  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/ 
Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
9490, 93sylib 196 . . . 4  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/)  \/  ( E. y  e.  On  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/  Lim  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
953, 64, 943syl 20 . . 3  |-  ( (
card `  A )  =  A  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  \/  ( E. y  e.  On  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/ 
Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
9695adantl 466 . 2  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
\/  ( E. y  e.  On  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  \/ 
Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
9729, 89, 96mpjaod 381 1  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   E.wrex 2710   {crab 2713    C_ wss 3321   (/)c0 3630   |^|cint 4121   U_ciun 4164   class class class wbr 4285   Ord word 4710   Oncon0 4711   Lim wlim 4712   suc csuc 4713   dom cdm 4832   ` cfv 5411   omcom 6471    ~<_ cdom 7300  harchar 7763   cardccrd 8097   alephcale 8098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-har 7765  df-card 8101  df-aleph 8102
This theorem is referenced by:  cardalephex  8252  tskcard  8940
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