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Theorem cardaleph 7716
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 7577 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2343 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 alephle 7715 . . . . . . . . 9  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
5 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
65sseq2d 3206 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  A ) ) )
76rspcev 2884 . . . . . . . . 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 2419 . . . . . . . . . 10  |-  F/_ x A
10 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x aleph
11 nfrab1 2720 . . . . . . . . . . . 12  |-  F/_ x { x  e.  On  |  A  C_  ( aleph `  x ) }
1211nfint 3872 . . . . . . . . . . 11  |-  F/_ x |^| { x  e.  On  |  A  C_  ( aleph `  x ) }
1310, 12nffv 5532 . . . . . . . . . 10  |-  F/_ x
( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
149, 13nfss 3173 . . . . . . . . 9  |-  F/ x  A  C_  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )
15 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  |^| { x  e.  On  |  A  C_  ( aleph `  x ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
1615sseq2d 3206 . . . . . . . . 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 4590 . . . . . . . 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 5525 . . . . . . . . 9  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  (
aleph `  (/) ) )
21 aleph0 7693 . . . . . . . . 9  |-  ( aleph `  (/) )  =  om
2220, 21syl6eq 2331 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  (/)  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  =  om )
2322sseq1d 3205 . . . . . . 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 3194 . . . . 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 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
3130sucid 4471 . . . . . . . . . . . 12  |-  y  e. 
suc  y
32 eleq2 2344 . . . . . . . . . . . 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 5525 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
3534sseq2d 3206 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( A  C_  ( aleph `  x
)  <->  A  C_  ( aleph `  y ) ) )
3635onnminsb 4595 . . . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  =  suc  y  ->  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x
) } )  =  ( aleph `  suc  y ) )
41 alephsuc 7695 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
4240, 41sylan9eqr 2337 . . . . . . . . . . 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 2350 . . . . . . . . . 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 7277 . . . . . . . . . . 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 7582 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  e.  dom  card )
483, 47syl 15 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  A  e. 
dom  card )
49 alephon 7696 . . . . . . . . . . . . 13  |-  ( aleph `  y )  e.  On
50 onenon 7582 . . . . . . . . . . . . 13  |-  ( (
aleph `  y )  e.  On  ->  ( aleph `  y )  e.  dom  card )
5149, 50ax-mp 8 . . . . . . . . . . . 12  |-  ( aleph `  y )  e.  dom  card
52 carddom2 7610 . . . . . . . . . . . 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 3199 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  ( card `  ( aleph `  y ) )  <-> 
A  C_  ( card `  ( aleph `  y )
) ) )
55 alephcard 7697 . . . . . . . . . . . . 13  |-  ( card `  ( aleph `  y )
)  =  ( aleph `  y )
5655sseq2i 3203 . . . . . . . . . . . 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 2666 . . . . 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 4593 . . . . . . . . . . . . . 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 4417 . . . . . . . . . . . . 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 4501 . . . . . . . . . . 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 2646 . . . . . . . . 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 7694 . . . . . . . . . . 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 2350 . . . . . . . . 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 3909 . . . . . . . . 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 7696 . . . . . . 7  |-  ( aleph ` 
|^| { x  e.  On  |  A  C_  ( aleph `  x ) } )  e.  On
85 onsseleq 4433 . . . . . . 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 4402 . . . . 5  |-  ( |^| { x  e.  On  |  A  C_  ( aleph `  x
) }  e.  On  ->  Ord  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } )
92 ordzsl 4636 . . . . . 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 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   |^|cint 3862   U_ciun 3905   class class class wbr 4023   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861  harchar 7270   cardccrd 7568   alephcale 7569
This theorem is referenced by:  cardalephex  7717  tskcard  8403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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