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Theorem cardaleph 7732
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 7593 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2356 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 alephle 7731 . . . . . . . . 9  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
5 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
65sseq2d 3219 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  A ) ) )
76rspcev 2897 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  C_  ( aleph `  A
) )  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
84, 7mpdan 649 . . . . . . . 8  |-  ( A  e.  On  ->  E. x  e.  On  A  C_  ( aleph `  x ) )
9 nfcv 2432 . . . . . . . . . 10  |-  F/_ x A
10 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x aleph
11 nfrab1 2733 . . . . . . . . . . . 12  |-  F/_ x { x  e.  On  |  A  C_  ( aleph `  x ) }
1211nfint 3888 . . . . . . . . . . 11  |-  F/_ x |^| { x  e.  On  |  A  C_  ( aleph `  x ) }
1310, 12nffv 5548 . . . . . . . . . 10  |-  F/_ x
( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
149, 13nfss 3186 . . . . . . . . 9  |-  F/ x  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
15 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1615sseq2d 3219 . . . . . . . . 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 4606 . . . . . . . 8  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  ->  A  C_  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
183, 8, 173syl 18 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1918a1i 10 . . . . . 6  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( card `  A
)  =  A  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
20 fveq2 5541 . . . . . . . . 9  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (
aleph `  (/) ) )
21 aleph0 7709 . . . . . . . . 9  |-  ( aleph `  (/) )  =  om
2220, 21syl6eq 2344 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  om )
2322sseq1d 3218 . . . . . . 7  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  C_  A 
<->  om  C_  A )
)
2423biimprd 214 . . . . . 6  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( om  C_  A  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  C_  A
) )
2519, 24anim12d 546 . . . . 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 3207 . . . . 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 218 . . . 4  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( ( ( card `  A )  =  A  /\  om  C_  A
)  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2827com12 27 . . 3  |-  ( ( ( card `  A
)  =  A  /\  om  C_  A )  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  (/) 
->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
2928ancoms 439 . 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 2804 . . . . . . . . . . . . 13  |-  y  e. 
_V
3130sucid 4487 . . . . . . . . . . . 12  |-  y  e. 
suc  y
32 eleq2 2357 . . . . . . . . . . . 12  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( y  e. 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  <->  y  e.  suc  y ) )
3331, 32mpbiri 224 . . . . . . . . . . 11  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
34 fveq2 5541 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
3534sseq2d 3219 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  y ) ) )
3635onnminsb 4611 . . . . . . . . . . 11  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  C_  ( aleph `  y ) ) )
3733, 36syl5 28 . . . . . . . . . 10  |-  ( y  e.  On  ->  ( |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y  ->  -.  A  C_  ( aleph `  y )
) )
3837imp 418 . . . . . . . . 9  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  -.  A  C_  ( aleph `  y
) )
3938adantl 452 . . . . . . . 8  |-  ( ( ( card `  A
)  =  A  /\  ( y  e.  On  /\ 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y ) )  ->  -.  A  C_  ( aleph `  y ) )
40 fveq2 5541 . . . . . . . . . . . 12  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  =  ( aleph `  suc  y ) )
41 alephsuc 7711 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
4240, 41sylan9eqr 2350 . . . . . . . . . . 11  |-  ( ( y  e.  On  /\  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  =  suc  y )  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (har
`  ( aleph `  y
) ) )
4342eleq2d 2363 . . . . . . . . . 10  |-  ( ( 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 198 . . . . . . . . 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
) ) ) )
45 elharval 7293 . . . . . . . . . . 11  |-  ( A  e.  (har `  ( aleph `  y ) )  <-> 
( A  e.  On  /\  A  ~<_  ( aleph `  y
) ) )
4645simprbi 450 . . . . . . . . . 10  |-  ( A  e.  (har `  ( aleph `  y ) )  ->  A  ~<_  ( aleph `  y ) )
47 onenon 7598 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  e.  dom  card )
483, 47syl 15 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  A  e. 
dom  card )
49 alephon 7712 . . . . . . . . . . . . 13  |-  ( aleph `  y )  e.  On
50 onenon 7598 . . . . . . . . . . . . 13  |-  ( (
aleph `  y )  e.  On  ->  ( aleph `  y )  e.  dom  card )
5149, 50ax-mp 8 . . . . . . . . . . . 12  |-  ( aleph `  y )  e.  dom  card
52 carddom2 7626 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  card  /\  ( aleph `  y )  e.  dom  card )  ->  (
( card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
5348, 51, 52sylancl 643 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  ~<_  ( aleph `  y
) ) )
54 sseq1 3212 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( card `  ( aleph `  y )
) ) )
55 alephcard 7713 . . . . . . . . . . . . 13  |-  ( card `  ( aleph `  y )
)  =  ( aleph `  y )
5655sseq2i 3216 . . . . . . . . . . . 12  |-  ( A 
C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) )
5754, 56syl6bb 252 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( aleph `  y ) ) )
5853, 57bitr3d 246 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  ->  ( A  ~<_  ( aleph `  y )  <->  A 
C_  ( aleph `  y
) ) )
5946, 58syl5ib 210 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( A  e.  (har `  ( aleph `  y ) )  ->  A  C_  ( aleph `  y ) ) )
6044, 59sylan9r 639 . . . . . . . 8  |-  ( ( ( 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 168 . . . . . . 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 ) } ) )
6261exp32 588 . . . . . 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 ) } ) ) ) )
6362rexlimdv 2679 . . . . 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 ) } ) ) )
64 onintrab2 4609 . . . . . . . . . . . . . 14  |-  ( E. x  e.  On  A  C_  ( aleph `  x )  <->  |^|
{ x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
658, 64sylib 188 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On )
66 onelon 4433 . . . . . . . . . . . . 13  |-  ( (
|^| { x  e.  On  |  A  C_  ( aleph `  x ) }  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  -> 
y  e.  On )
6765, 66sylan 457 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  y  e.  On )
6836adantld 453 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  (
( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) ) )
6967, 68mpcom 32 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  C_  ( aleph `  y
) )
7049onelssi 4517 . . . . . . . . . . 11  |-  ( A  e.  ( aleph `  y
)  ->  A  C_  ( aleph `  y ) )
7169, 70nsyl 113 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph `  y ) )
7271nrexdv 2659 . . . . . . . . 9  |-  ( A  e.  On  ->  -.  E. y  e.  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } A  e.  ( aleph `  y ) )
7372adantr 451 . . . . . . . 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 ) )
74 alephlim 7710 . . . . . . . . . . 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
) )
7565, 74sylan 457 . . . . . . . . . 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
) )
7675eleq2d 2363 . . . . . . . . 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
) ) )
77 eliun 3925 . . . . . . . . 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 ) )
7876, 77syl6bb 252 . . . . . . . 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 ) ) )
7973, 78mtbird 292 . . . . . . 7  |-  ( ( A  e.  On  /\  Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
8079ex 423 . . . . . 6  |-  ( A  e.  On  ->  ( Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
813, 80syl 15 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( Lim  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  -.  A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8263, 81jaod 369 . . . 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 ) } ) ) )
838, 17syl 15 . . . . . 6  |-  ( A  e.  On  ->  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
84 alephon 7712 . . . . . . 7  |-  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On
85 onsseleq 4449 . . . . . . 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
) } ) ) ) )
8684, 85mpan2 652 . . . . . 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
) } ) ) ) )
8783, 86mpbid 201 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  \/  A  =  (
aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) ) )
8887ord 366 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } ) ) )
893, 82, 88sylsyld 52 . . 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 ) } ) ) )
9089adantl 452 . 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
) } ) ) )
91 eloni 4418 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
92 ordzsl 4652 . . . . . 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 ) } ) )
93 3orass 937 . . . . . 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 ) } ) ) )
9492, 93bitri 240 . . . . 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 ) } ) ) )
9591, 94sylib 188 . . . 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 ) } ) ) )
963, 65, 953syl 18 . . 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 ) } ) ) )
9796adantl 452 . 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 ) } ) ) )
9829, 90, 97mpjaod 370 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   |^|cint 3878   U_ciun 3921   class class class wbr 4039   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410   omcom 4672   dom cdm 4705   ` cfv 5271    ~<_ cdom 6877  harchar 7286   cardccrd 7584   alephcale 7585
This theorem is referenced by:  cardalephex  7733  tskcard  8419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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