MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardaleph Structured version   Unicode version

Theorem cardaleph 8383
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 8238 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2454 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 alephle 8382 . . . . . . . . 9  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
5 fveq2 5774 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
65sseq2d 3445 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  A ) ) )
76rspcev 3135 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  C_  ( aleph `  A
) )  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
84, 7mpdan 666 . . . . . . . 8  |-  ( A  e.  On  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
9 nfcv 2544 . . . . . . . . . 10  |-  F/_ x A
10 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x aleph
11 nfrab1 2963 . . . . . . . . . . . 12  |-  F/_ x { x  e.  On  |  A  C_  ( aleph `  x ) }
1211nfint 4209 . . . . . . . . . . 11  |-  F/_ x |^| { x  e.  On  |  A  C_  ( aleph `  x ) }
1310, 12nffv 5781 . . . . . . . . . 10  |-  F/_ x
( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
149, 13nfss 3410 . . . . . . . . 9  |-  F/ x  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
15 fveq2 5774 . . . . . . . . . 10  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1615sseq2d 3445 . . . . . . . . 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 6533 . . . . . . . 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 5774 . . . . . . . . 9  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (
aleph `  (/) ) )
21 aleph0 8360 . . . . . . . . 9  |-  ( aleph `  (/) )  =  om
2220, 21syl6eq 2439 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  om )
2322sseq1d 3444 . . . . . . 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 561 . . . . 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 3432 . . . . 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 451 . 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 3037 . . . . . . . . . . . 12  |-  y  e. 
_V
3130sucid 4871 . . . . . . . . . . 11  |-  y  e. 
suc  y
32 eleq2 2455 . . . . . . . . . . 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 5774 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
3534sseq2d 3445 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  y ) ) )
3635onnminsb 6538 . . . . . . . . . 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 427 . . . . . . . 8  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  -.  A  C_  ( aleph `  y
) )
3938adantl 464 . . . . . . 7  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  ->  -.  A  C_  ( aleph `  y ) )
40 fveq2 5774 . . . . . . . . . . 11  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  =  ( aleph `  suc  y ) )
41 alephsuc 8362 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
4240, 41sylan9eqr 2445 . . . . . . . . . 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 2452 . . . . . . . . 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 7904 . . . . . . . . . 10  |-  ( A  e.  (har `  ( aleph `  y ) )  <-> 
( A  e.  On  /\  A  ~<_  ( aleph `  y
) ) )
4645simprbi 462 . . . . . . . . 9  |-  ( A  e.  (har `  ( aleph `  y ) )  ->  A  ~<_  ( aleph `  y ) )
47 onenon 8243 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  A  e.  dom  card )
483, 47syl 16 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  A  e. 
dom  card )
49 alephon 8363 . . . . . . . . . . . 12  |-  ( aleph `  y )  e.  On
50 onenon 8243 . . . . . . . . . . . 12  |-  ( (
aleph `  y )  e.  On  ->  ( aleph `  y )  e.  dom  card )
5149, 50ax-mp 5 . . . . . . . . . . 11  |-  ( aleph `  y )  e.  dom  card
52 carddom2 8271 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\  ( aleph `  y )  e.  dom  card )  ->  (
( card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
5348, 51, 52sylancl 660 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
54 sseq1 3438 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( card `  ( aleph `  y )
) ) )
55 alephcard 8364 . . . . . . . . . . . 12  |-  ( card `  ( aleph `  y )
)  =  ( aleph `  y )
5655sseq2i 3442 . . . . . . . . . . 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 656 . . . . . . 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 2875 . . . . 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 6536 . . . . . . . . . . . . . 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 4817 . . . . . . . . . . . . 13  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  -> 
y  e.  On )
6664, 65sylan 469 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  y  e.  On )
6736adantld 465 . . . . . . . . . . . 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 4900 . . . . . . . . . . 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 2838 . . . . . . . . 9  |-  ( A  e.  On  ->  -.  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
7271adantr 463 . . . . . . . 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 8361 . . . . . . . . . . 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 469 . . . . . . . . . 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 2452 . . . . . . . . 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 4248 . . . . . . . . 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 299 . . . . . . 7  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
7978ex 432 . . . . . 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 378 . . . 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 8363 . . . . . . 7  |-  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On
84 onsseleq 4833 . . . . . . 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 669 . . . . . 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 375 . . . 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 464 . 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 4802 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
91 ordzsl 6579 . . . . . 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 974 . . . . . 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 464 . 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 379 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 366    /\ wa 367    \/ w3o 970    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736    C_ wss 3389   (/)c0 3711   |^|cint 4199   U_ciun 4243   class class class wbr 4367   Ord word 4791   Oncon0 4792   Lim wlim 4793   suc csuc 4794   dom cdm 4913   ` cfv 5496   omcom 6599    ~<_ cdom 7433  harchar 7897   cardccrd 8229   alephcale 8230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-oi 7850  df-har 7899  df-card 8233  df-aleph 8234
This theorem is referenced by:  cardalephex  8384  tskcard  9070
  Copyright terms: Public domain W3C validator