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Theorem alephval3 7980
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 7940 . . . 4  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
21a1i 11 . . 3  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
3 alephgeom 7952 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
43biimpi 187 . . 3  |-  ( A  e.  On  ->  om  C_  ( aleph `  A ) )
5 alephord2i 7947 . . . . 5  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
6 elirr 7555 . . . . . . 7  |-  -.  ( aleph `  y )  e.  ( aleph `  y )
7 eleq2 2496 . . . . . . 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 2780 . . 3  |-  ( A  e.  On  ->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
)
12 fvex 5733 . . . 4  |-  ( aleph `  A )  e.  _V
13 fveq2 5719 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( card `  x )  =  (
card `  ( aleph `  A
) ) )
14 id 20 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  x  =  ( aleph `  A )
)
1513, 14eqeq12d 2449 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( ( card `  x )  =  x  <->  ( card `  ( aleph `  A ) )  =  ( aleph `  A
) ) )
16 sseq2 3362 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( om  C_  x  <->  om  C_  ( aleph `  A ) ) )
17 eqeq1 2441 . . . . . . 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 2717 . . . . 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 3074 . . 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 7960 . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( aleph `  y
)  ->  ( z  e.  ( aleph `  A )  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
26 alephord2 7946 . . . . . . . . . . . . . . . . 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 2807 . . . . . . . . 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 2710 . . . . . . 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 2951 . . . . . . 7  |-  z  e. 
_V
45 fveq2 5719 . . . . . . . . 9  |-  ( x  =  z  ->  ( card `  x )  =  ( card `  z
) )
46 id 20 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
4745, 46eqeq12d 2449 . . . . . . . 8  |-  ( x  =  z  ->  (
( card `  x )  =  x  <->  ( card `  z
)  =  z ) )
48 sseq2 3362 . . . . . . . 8  |-  ( x  =  z  ->  ( om  C_  x  <->  om  C_  z
) )
49 eqeq1 2441 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  =  ( aleph `  y )  <->  z  =  ( aleph `  y )
) )
5049notbid 286 . . . . . . . . 9  |-  ( x  =  z  ->  ( -.  x  =  ( aleph `  y )  <->  -.  z  =  ( aleph `  y
) ) )
5150ralbidv 2717 . . . . . . . 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 3074 . . . . . 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 2780 . 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 7820 . . . . . 6  |-  ( card `  x )  e.  On
60 eleq1 2495 . . . . . 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 3410 . . 3  |-  { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y ) ) }  C_  On
64 oneqmini 4624 . . 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 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698    C_ wss 3312   |^|cint 4042   Oncon0 4573   omcom 4836   ` cfv 5445   cardccrd 7811   alephcale 7812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-reg 7549  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816
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