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Theorem alephval3 7947
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 7907 . . . 4  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
21a1i 11 . . 3  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
3 alephgeom 7919 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
43biimpi 187 . . 3  |-  ( A  e.  On  ->  om  C_  ( aleph `  A ) )
5 alephord2i 7914 . . . . 5  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
6 elirr 7522 . . . . . . 7  |-  -.  ( aleph `  y )  e.  ( aleph `  y )
7 eleq2 2465 . . . . . . 7  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  ( aleph `  y )  e.  (
aleph `  y ) ) )
86, 7mtbiri 295 . . . . . 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 2748 . . 3  |-  ( A  e.  On  ->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
)
12 fvex 5701 . . . 4  |-  ( aleph `  A )  e.  _V
13 fveq2 5687 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( card `  x )  =  (
card `  ( aleph `  A
) ) )
14 id 20 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  x  =  ( aleph `  A )
)
1513, 14eqeq12d 2418 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( ( card `  x )  =  x  <->  ( card `  ( aleph `  A ) )  =  ( aleph `  A
) ) )
16 sseq2 3330 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( om  C_  x  <->  om  C_  ( aleph `  A ) ) )
17 eqeq1 2410 . . . . . . 7  |-  ( x  =  ( aleph `  A
)  ->  ( x  =  ( aleph `  y
)  <->  ( aleph `  A
)  =  ( aleph `  y ) ) )
1817notbid 286 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( -.  x  =  ( aleph `  y )  <->  -.  ( aleph `  A )  =  ( aleph `  y )
) )
1918ralbidv 2686 . . . . 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 3042 . . 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 7927 . . . . . . . . . 10  |-  ( om  C_  z  ->  ( (
card `  z )  =  z  <->  E. y  e.  On  z  =  ( aleph `  y ) ) )
2423biimpac 473 . . . . . . . . 9  |-  ( ( ( card `  z
)  =  z  /\  om  C_  z )  ->  E. y  e.  On  z  =  (
aleph `  y ) )
25 eleq1 2464 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( aleph `  y
)  ->  ( z  e.  ( aleph `  A )  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
26 alephord2 7913 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  e.  A  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
2726bicomd 193 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  y  e.  A ) )
2825, 27sylan9bbr 682 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( z  e.  ( aleph `  A )  <->  y  e.  A ) )
2928biimpcd 216 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  y  e.  A
) )
30 simpr 448 . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . 13  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( y  e.  A  /\  z  =  ( aleph `  y )
) ) )
3332exp4c 592 . . . . . . . . . . . 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 574 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( y  e.  On  /\  z  =  ( aleph `  y )
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) )
3635reximdv2 2775 . . . . . . . . 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 419 . . . . . . 7  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  E. y  e.  A  z  =  ( aleph `  y ) )
39 dfrex2 2679 . . . . . . 7  |-  ( E. y  e.  A  z  =  ( aleph `  y
)  <->  -.  A. y  e.  A  -.  z  =  ( aleph `  y
) )
4038, 39sylib 189 . . . . . 6  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  -.  A. y  e.  A  -.  z  =  ( aleph `  y ) )
41 nan 564 . . . . . 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 201 . . . . 5  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
4342ex 424 . . . 4  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) ) )
44 vex 2919 . . . . . . 7  |-  z  e. 
_V
45 fveq2 5687 . . . . . . . . 9  |-  ( x  =  z  ->  ( card `  x )  =  ( card `  z
) )
46 id 20 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
4745, 46eqeq12d 2418 . . . . . . . 8  |-  ( x  =  z  ->  (
( card `  x )  =  x  <->  ( card `  z
)  =  z ) )
48 sseq2 3330 . . . . . . . 8  |-  ( x  =  z  ->  ( om  C_  x  <->  om  C_  z
) )
49 eqeq1 2410 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  =  ( aleph `  y )  <->  z  =  ( aleph `  y )
) )
5049notbid 286 . . . . . . . . 9  |-  ( x  =  z  ->  ( -.  x  =  ( aleph `  y )  <->  -.  z  =  ( aleph `  y
) ) )
5150ralbidv 2686 . . . . . . . 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 3042 . . . . . 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 241 . . . . 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 288 . . . 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 219 . . 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 2748 . 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 7787 . . . . . 6  |-  ( card `  x )  e.  On
60 eleq1 2464 . . . . . 6  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
6159, 60mpbii 203 . . . . 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 3378 . . 3  |-  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  C_  On
64 oneqmini 4592 . . 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 8 . 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 643 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    C_ wss 3280   |^|cint 4010   Oncon0 4541   omcom 4804   ` cfv 5413   cardccrd 7778   alephcale 7779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783
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