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Theorem cardaleph 7972
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 7833 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2498 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 204 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 alephle 7971 . . . . . . . . 9  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
5 fveq2 5730 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
65sseq2d 3378 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  A ) ) )
76rspcev 3054 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  C_  ( aleph `  A
) )  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
84, 7mpdan 651 . . . . . . . 8  |-  ( A  e.  On  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
9 nfcv 2574 . . . . . . . . . 10  |-  F/_ x A
10 nfcv 2574 . . . . . . . . . . 11  |-  F/_ x aleph
11 nfrab1 2890 . . . . . . . . . . . 12  |-  F/_ x { x  e.  On  |  A  C_  ( aleph `  x ) }
1211nfint 4062 . . . . . . . . . . 11  |-  F/_ x |^| { x  e.  On  |  A  C_  ( aleph `  x ) }
1310, 12nffv 5737 . . . . . . . . . 10  |-  F/_ x
( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
149, 13nfss 3343 . . . . . . . . 9  |-  F/ x  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
15 fveq2 5730 . . . . . . . . . 10  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1615sseq2d 3378 . . . . . . . . 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 4781 . . . . . . . 8  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  ->  A  C_  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
183, 8, 173syl 19 . . . . . . 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 5730 . . . . . . . . 9  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (
aleph `  (/) ) )
21 aleph0 7949 . . . . . . . . 9  |-  ( aleph `  (/) )  =  om
2220, 21syl6eq 2486 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  om )
2322sseq1d 3377 . . . . . . 7  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  C_  A 
<->  om  C_  A )
)
2423biimprd 216 . . . . . 6  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( om  C_  A  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  C_  A
) )
2519, 24anim12d 548 . . . . 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 3365 . . . . 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 220 . . . 4  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( ( card `  A )  =  A  /\  om  C_  A
)  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2827com12 30 . . 3  |-  ( ( ( card `  A
)  =  A  /\  om  C_  A )  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2928ancoms 441 . 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 2961 . . . . . . . . . . . 12  |-  y  e. 
_V
3130sucid 4662 . . . . . . . . . . 11  |-  y  e. 
suc  y
32 eleq2 2499 . . . . . . . . . . 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 226 . . . . . . . . . 10  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
34 fveq2 5730 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
3534sseq2d 3378 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  y ) ) )
3635onnminsb 4786 . . . . . . . . . 10  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  C_  ( aleph `  y ) ) )
3733, 36syl5 31 . . . . . . . . 9  |-  ( y  e.  On  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  ->  -.  A  C_  ( aleph `  y )
) )
3837imp 420 . . . . . . . 8  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  -.  A  C_  ( aleph `  y
) )
3938adantl 454 . . . . . . 7  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  ->  -.  A  C_  ( aleph `  y ) )
40 fveq2 5730 . . . . . . . . . . 11  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  =  ( aleph `  suc  y ) )
41 alephsuc 7951 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
4240, 41sylan9eqr 2492 . . . . . . . . . 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 2505 . . . . . . . . 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 200 . . . . . . . 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 7533 . . . . . . . . . 10  |-  ( A  e.  (har `  ( aleph `  y ) )  <-> 
( A  e.  On  /\  A  ~<_  ( aleph `  y
) ) )
4645simprbi 452 . . . . . . . . 9  |-  ( A  e.  (har `  ( aleph `  y ) )  ->  A  ~<_  ( aleph `  y ) )
47 onenon 7838 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  A  e.  dom  card )
483, 47syl 16 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  A  e. 
dom  card )
49 alephon 7952 . . . . . . . . . . . 12  |-  ( aleph `  y )  e.  On
50 onenon 7838 . . . . . . . . . . . 12  |-  ( (
aleph `  y )  e.  On  ->  ( aleph `  y )  e.  dom  card )
5149, 50ax-mp 8 . . . . . . . . . . 11  |-  ( aleph `  y )  e.  dom  card
52 carddom2 7866 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\  ( aleph `  y )  e.  dom  card )  ->  (
( card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
5348, 51, 52sylancl 645 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
54 sseq1 3371 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( card `  ( aleph `  y )
) ) )
55 alephcard 7953 . . . . . . . . . . . 12  |-  ( card `  ( aleph `  y )
)  =  ( aleph `  y )
5655sseq2i 3375 . . . . . . . . . . 11  |-  ( A 
C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) )
5754, 56syl6bb 254 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) ) )
5853, 57bitr3d 248 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( A  ~<_  ( aleph `  y )  <->  A 
C_  ( aleph `  y
) ) )
5946, 58syl5ib 212 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  ( A  e.  (har `  ( aleph `  y ) )  ->  A  C_  ( aleph `  y ) ) )
6044, 59sylan9r 641 . . . . . . 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 171 . . . . . 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 2833 . . . . 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 4784 . . . . . . . . . . . . . 14  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  <->  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
648, 63sylib 190 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
65 onelon 4608 . . . . . . . . . . . . 13  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  -> 
y  e.  On )
6664, 65sylan 459 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  y  e.  On )
6736adantld 455 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  (
( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) ) )
6866, 67mpcom 35 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) )
6949onelssi 4692 . . . . . . . . . . 11  |-  ( A  e.  ( aleph `  y
)  ->  A  C_  ( aleph `  y ) )
7068, 69nsyl 116 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph `  y ) )
7170nrexdv 2811 . . . . . . . . 9  |-  ( A  e.  On  ->  -.  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
7271adantr 453 . . . . . . . 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 7950 . . . . . . . . . . 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 459 . . . . . . . . . 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 2505 . . . . . . . . 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 4099 . . . . . . . . 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 254 . . . . . . . 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 294 . . . . . . 7  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
7978ex 425 . . . . . 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 371 . . . 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 7952 . . . . . . 7  |-  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On
84 onsseleq 4624 . . . . . . 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 654 . . . . . 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 203 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  \/  A  =  (
aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8786ord 368 . . . 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 55 . . 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 454 . 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 4593 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
91 ordzsl 4827 . . . . . 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 940 . . . . . 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 242 . . . . 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 190 . . . 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 19 . . 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 454 . 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 372 1  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   |^|cint 4052   U_ciun 4095   class class class wbr 4214   Ord word 4582   Oncon0 4583   Lim wlim 4584   suc csuc 4585   omcom 4847   dom cdm 4880   ` cfv 5456    ~<_ cdom 7109  harchar 7526   cardccrd 7824   alephcale 7825
This theorem is referenced by:  cardalephex  7973  tskcard  8658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-har 7528  df-card 7828  df-aleph 7829
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