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Theorem alephval3 7733
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
Dummy variable  z is distinct from all other variables.

Proof of Theorem alephval3
StepHypRef Expression
1 alephcard 7693 . . . 4  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
21a1i 12 . . 3  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
3 alephgeom 7705 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
43biimpi 188 . . 3  |-  ( A  e.  On  ->  om  C_  ( aleph `  A ) )
5 alephord2i 7700 . . . . 5  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
6 elirr 7308 . . . . . . 7  |-  -.  ( aleph `  y )  e.  ( aleph `  y )
7 eleq2 2346 . . . . . . 7  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  ( aleph `  y )  e.  (
aleph `  y ) ) )
86, 7mtbiri 296 . . . . . 6  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  -.  ( aleph `  y
)  e.  ( aleph `  A ) )
98con2i 114 . . . . 5  |-  ( (
aleph `  y )  e.  ( aleph `  A )  ->  -.  ( aleph `  A
)  =  ( aleph `  y ) )
105, 9syl6 31 . . . 4  |-  ( A  e.  On  ->  (
y  e.  A  ->  -.  ( aleph `  A )  =  ( aleph `  y
) ) )
1110ralrimiv 2627 . . 3  |-  ( A  e.  On  ->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
)
12 fvex 5500 . . . 4  |-  ( aleph `  A )  e.  _V
13 fveq2 5486 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( card `  x )  =  (
card `  ( aleph `  A
) ) )
14 id 21 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  x  =  ( aleph `  A )
)
1513, 14eqeq12d 2299 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( ( card `  x )  =  x  <->  ( card `  ( aleph `  A ) )  =  ( aleph `  A
) ) )
16 sseq2 3202 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( om  C_  x  <->  om  C_  ( aleph `  A ) ) )
17 eqeq1 2291 . . . . . . 7  |-  ( x  =  ( aleph `  A
)  ->  ( x  =  ( aleph `  y
)  <->  ( aleph `  A
)  =  ( aleph `  y ) ) )
1817notbid 287 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( -.  x  =  ( aleph `  y )  <->  -.  ( aleph `  A )  =  ( aleph `  y )
) )
1918ralbidv 2565 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
) )
2015, 16, 193anbi123d 1254 . . . 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 2916 . . 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 1138 . 2  |-  ( A  e.  On  ->  ( aleph `  A )  e. 
{ x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
23 cardalephex 7713 . . . . . . . . . 10  |-  ( om  C_  z  ->  ( (
card `  z )  =  z  <->  E. y  e.  On  z  =  ( aleph `  y ) ) )
2423biimpac 474 . . . . . . . . 9  |-  ( ( ( card `  z
)  =  z  /\  om  C_  z )  ->  E. y  e.  On  z  =  (
aleph `  y ) )
25 eleq1 2345 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( aleph `  y
)  ->  ( z  e.  ( aleph `  A )  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
26 alephord2 7699 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  e.  A  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
2726bicomd 194 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  y  e.  A ) )
2825, 27sylan9bbr 683 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( z  e.  ( aleph `  A )  <->  y  e.  A ) )
2928biimpcd 217 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  y  e.  A
) )
30 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) )
3130a1i 12 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) ) )
3229, 31jcad 521 . . . . . . . . . . . . 13  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( y  e.  A  /\  z  =  ( aleph `  y )
) ) )
3332exp4c 593 . . . . . . . . . . . 12  |-  ( z  e.  ( aleph `  A
)  ->  ( y  e.  On  ->  ( A  e.  On  ->  ( z  =  ( aleph `  y
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) ) ) )
3433com3r 75 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  (
y  e.  On  ->  ( z  =  ( aleph `  y )  ->  (
y  e.  A  /\  z  =  ( aleph `  y ) ) ) ) ) )
3534imp4b 575 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( y  e.  On  /\  z  =  ( aleph `  y )
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) )
3635reximdv2 2654 . . . . . . . . 9  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( E. y  e.  On  z  =  (
aleph `  y )  ->  E. y  e.  A  z  =  ( aleph `  y ) ) )
3724, 36syl5 30 . . . . . . . 8  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( ( card `  z )  =  z  /\  om  C_  z
)  ->  E. y  e.  A  z  =  ( aleph `  y )
) )
3837imp 420 . . . . . . 7  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  E. y  e.  A  z  =  ( aleph `  y ) )
39 dfrex2 2558 . . . . . . 7  |-  ( E. y  e.  A  z  =  ( aleph `  y
)  <->  -.  A. y  e.  A  -.  z  =  ( aleph `  y
) )
4038, 39sylib 190 . . . . . 6  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  -.  A. y  e.  A  -.  z  =  ( aleph `  y ) )
41 nan 565 . . . . . 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 202 . . . . 5  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
4342ex 425 . . . 4  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) ) )
44 vex 2793 . . . . . . 7  |-  z  e. 
_V
45 fveq2 5486 . . . . . . . . 9  |-  ( x  =  z  ->  ( card `  x )  =  ( card `  z
) )
46 id 21 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
4745, 46eqeq12d 2299 . . . . . . . 8  |-  ( x  =  z  ->  (
( card `  x )  =  x  <->  ( card `  z
)  =  z ) )
48 sseq2 3202 . . . . . . . 8  |-  ( x  =  z  ->  ( om  C_  x  <->  om  C_  z
) )
49 eqeq1 2291 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  =  ( aleph `  y )  <->  z  =  ( aleph `  y )
) )
5049notbid 287 . . . . . . . . 9  |-  ( x  =  z  ->  ( -.  x  =  ( aleph `  y )  <->  -.  z  =  ( aleph `  y
) ) )
5150ralbidv 2565 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5247, 48, 513anbi123d 1254 . . . . . . 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 2916 . . . . . 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 938 . . . . . 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 242 . . . . 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 289 . . . 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 220 . . 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 2627 . 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 7573 . . . . . 6  |-  ( card `  x )  e.  On
60 eleq1 2345 . . . . . 6  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
6159, 60mpbii 204 . . . . 5  |-  ( (
card `  x )  =  x  ->  x  e.  On )
62613ad2ant1 978 . . . 4  |-  ( ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) )  ->  x  e.  On )
6362abssi 3250 . . 3  |-  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  C_  On
64 oneqmini 4443 . . 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 10 . 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 644 1  |-  ( A  e.  On  ->  ( aleph `  A )  = 
|^| { x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   {cab 2271   A.wral 2545   E.wrex 2546    C_ wss 3154   |^|cint 3864   Oncon0 4392   omcom 4656   ` cfv 5222   cardccrd 7564   alephcale 7565
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7302  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-har 7268  df-card 7568  df-aleph 7569
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