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Theorem alephval3 8559
Description: An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
Assertion
Ref Expression
alephval3  |-  ( A  e.  On  ->  ( aleph `  A )  = 
|^| { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
Distinct variable group:    x, y, A

Proof of Theorem alephval3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 alephcard 8519 . . . 4  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
21a1i 11 . . 3  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
3 alephgeom 8531 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
43biimpi 199 . . 3  |-  ( A  e.  On  ->  om  C_  ( aleph `  A ) )
5 alephord2i 8526 . . . . 5  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
6 elirr 8131 . . . . . . 7  |-  -.  ( aleph `  y )  e.  ( aleph `  y )
7 eleq2 2538 . . . . . . 7  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  ( aleph `  y )  e.  (
aleph `  y ) ) )
86, 7mtbiri 310 . . . . . 6  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  -.  ( aleph `  y
)  e.  ( aleph `  A ) )
98con2i 124 . . . . 5  |-  ( (
aleph `  y )  e.  ( aleph `  A )  ->  -.  ( aleph `  A
)  =  ( aleph `  y ) )
105, 9syl6 33 . . . 4  |-  ( A  e.  On  ->  (
y  e.  A  ->  -.  ( aleph `  A )  =  ( aleph `  y
) ) )
1110ralrimiv 2808 . . 3  |-  ( A  e.  On  ->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
)
12 fvex 5889 . . . 4  |-  ( aleph `  A )  e.  _V
13 fveq2 5879 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( card `  x )  =  (
card `  ( aleph `  A
) ) )
14 id 22 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  x  =  ( aleph `  A )
)
1513, 14eqeq12d 2486 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( ( card `  x )  =  x  <->  ( card `  ( aleph `  A ) )  =  ( aleph `  A
) ) )
16 sseq2 3440 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( om  C_  x  <->  om  C_  ( aleph `  A ) ) )
17 eqeq1 2475 . . . . . . 7  |-  ( x  =  ( aleph `  A
)  ->  ( x  =  ( aleph `  y
)  <->  ( aleph `  A
)  =  ( aleph `  y ) ) )
1817notbid 301 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( -.  x  =  ( aleph `  y )  <->  -.  ( aleph `  A )  =  ( aleph `  y )
) )
1918ralbidv 2829 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
) )
2015, 16, 193anbi123d 1365 . . . 4  |-  ( x  =  ( aleph `  A
)  ->  ( (
( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) )  <-> 
( ( card `  ( aleph `  A ) )  =  ( aleph `  A
)  /\  om  C_  ( aleph `  A )  /\  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y
) ) ) )
2112, 20elab 3173 . . 3  |-  ( (
aleph `  A )  e. 
{ x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) }  <->  ( ( card `  ( aleph `  A
) )  =  (
aleph `  A )  /\  om  C_  ( aleph `  A )  /\  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y
) ) )
222, 4, 11, 21syl3anbrc 1214 . 2  |-  ( A  e.  On  ->  ( aleph `  A )  e. 
{ x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
23 cardalephex 8539 . . . . . . . . . 10  |-  ( om  C_  z  ->  ( (
card `  z )  =  z  <->  E. y  e.  On  z  =  ( aleph `  y ) ) )
2423biimpac 494 . . . . . . . . 9  |-  ( ( ( card `  z
)  =  z  /\  om  C_  z )  ->  E. y  e.  On  z  =  (
aleph `  y ) )
25 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( aleph `  y
)  ->  ( z  e.  ( aleph `  A )  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
26 alephord2 8525 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  e.  A  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
2726bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  y  e.  A ) )
2825, 27sylan9bbr 715 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( z  e.  ( aleph `  A )  <->  y  e.  A ) )
2928biimpcd 232 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  y  e.  A
) )
30 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) )
3130a1i 11 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) ) )
3229, 31jcad 542 . . . . . . . . . . . . 13  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( y  e.  A  /\  z  =  ( aleph `  y )
) ) )
3332exp4c 619 . . . . . . . . . . . 12  |-  ( z  e.  ( aleph `  A
)  ->  ( y  e.  On  ->  ( A  e.  On  ->  ( z  =  ( aleph `  y
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) ) ) )
3433com3r 81 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  (
y  e.  On  ->  ( z  =  ( aleph `  y )  ->  (
y  e.  A  /\  z  =  ( aleph `  y ) ) ) ) ) )
3534imp4b 601 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( y  e.  On  /\  z  =  ( aleph `  y )
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) )
3635reximdv2 2855 . . . . . . . . 9  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( E. y  e.  On  z  =  (
aleph `  y )  ->  E. y  e.  A  z  =  ( aleph `  y ) ) )
3724, 36syl5 32 . . . . . . . 8  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( ( card `  z )  =  z  /\  om  C_  z
)  ->  E. y  e.  A  z  =  ( aleph `  y )
) )
3837imp 436 . . . . . . 7  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  E. y  e.  A  z  =  ( aleph `  y ) )
39 dfrex2 2837 . . . . . . 7  |-  ( E. y  e.  A  z  =  ( aleph `  y
)  <->  -.  A. y  e.  A  -.  z  =  ( aleph `  y
) )
4038, 39sylib 201 . . . . . 6  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  -.  A. y  e.  A  -.  z  =  ( aleph `  y ) )
41 nan 590 . . . . . 6  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )  <->  ( (
( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  -.  A. y  e.  A  -.  z  =  ( aleph `  y ) ) )
4240, 41mpbir 214 . . . . 5  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
4342ex 441 . . . 4  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) ) )
44 vex 3034 . . . . . . 7  |-  z  e. 
_V
45 fveq2 5879 . . . . . . . . 9  |-  ( x  =  z  ->  ( card `  x )  =  ( card `  z
) )
46 id 22 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
4745, 46eqeq12d 2486 . . . . . . . 8  |-  ( x  =  z  ->  (
( card `  x )  =  x  <->  ( card `  z
)  =  z ) )
48 sseq2 3440 . . . . . . . 8  |-  ( x  =  z  ->  ( om  C_  x  <->  om  C_  z
) )
49 eqeq1 2475 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  =  ( aleph `  y )  <->  z  =  ( aleph `  y )
) )
5049notbid 301 . . . . . . . . 9  |-  ( x  =  z  ->  ( -.  x  =  ( aleph `  y )  <->  -.  z  =  ( aleph `  y
) ) )
5150ralbidv 2829 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5247, 48, 513anbi123d 1365 . . . . . . 7  |-  ( x  =  z  ->  (
( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) )  <->  ( ( card `  z )  =  z  /\  om  C_  z  /\  A. y  e.  A  -.  z  =  ( aleph `  y ) ) ) )
5344, 52elab 3173 . . . . . 6  |-  ( z  e.  { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) }  <->  ( ( card `  z )  =  z  /\  om  C_  z  /\  A. y  e.  A  -.  z  =  ( aleph `  y ) ) )
54 df-3an 1009 . . . . . 6  |-  ( ( ( card `  z
)  =  z  /\  om  C_  z  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) )  <->  ( (
( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5553, 54bitri 257 . . . . 5  |-  ( z  e.  { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) }  <->  ( (
( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5655notbii 303 . . . 4  |-  ( -.  z  e.  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  <->  -.  ( (
( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5743, 56syl6ibr 235 . . 3  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  -.  z  e.  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) } ) )
5857ralrimiv 2808 . 2  |-  ( A  e.  On  ->  A. z  e.  ( aleph `  A )  -.  z  e.  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) } )
59 cardon 8396 . . . . . 6  |-  ( card `  x )  e.  On
60 eleq1 2537 . . . . . 6  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
6159, 60mpbii 216 . . . . 5  |-  ( (
card `  x )  =  x  ->  x  e.  On )
62613ad2ant1 1051 . . . 4  |-  ( ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) )  ->  x  e.  On )
6362abssi 3490 . . 3  |-  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  C_  On
64 oneqmini 5481 . . 3  |-  ( { x  |  ( (
card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  C_  On  ->  ( ( ( aleph `  A
)  e.  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  /\  A. z  e.  ( aleph `  A )  -.  z  e.  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) } )  ->  ( aleph `  A )  = 
|^| { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } ) )
6563, 64ax-mp 5 . 2  |-  ( ( ( aleph `  A )  e.  { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) }  /\  A. z  e.  ( aleph `  A )  -.  z  e.  { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )  ->  ( aleph `  A
)  =  |^| { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) } )
6622, 58, 65syl2anc 673 1  |-  ( A  e.  On  ->  ( aleph `  A )  = 
|^| { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757    C_ wss 3390   |^|cint 4226   Oncon0 5430   ` cfv 5589   omcom 6711   cardccrd 8387   alephcale 8388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392
This theorem is referenced by: (None)
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