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Theorem alephval3 7737
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 7697 . . . 4  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
21a1i 10 . . 3  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
3 alephgeom 7709 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
43biimpi 186 . . 3  |-  ( A  e.  On  ->  om  C_  ( aleph `  A ) )
5 alephord2i 7704 . . . . 5  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
6 elirr 7312 . . . . . . 7  |-  -.  ( aleph `  y )  e.  ( aleph `  y )
7 eleq2 2344 . . . . . . 7  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  ( aleph `  y )  e.  (
aleph `  y ) ) )
86, 7mtbiri 294 . . . . . 6  |-  ( (
aleph `  A )  =  ( aleph `  y )  ->  -.  ( aleph `  y
)  e.  ( aleph `  A ) )
98con2i 112 . . . . 5  |-  ( (
aleph `  y )  e.  ( aleph `  A )  ->  -.  ( aleph `  A
)  =  ( aleph `  y ) )
105, 9syl6 29 . . . 4  |-  ( A  e.  On  ->  (
y  e.  A  ->  -.  ( aleph `  A )  =  ( aleph `  y
) ) )
1110ralrimiv 2625 . . 3  |-  ( A  e.  On  ->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
)
12 fvex 5539 . . . 4  |-  ( aleph `  A )  e.  _V
13 fveq2 5525 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( card `  x )  =  (
card `  ( aleph `  A
) ) )
14 id 19 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  x  =  ( aleph `  A )
)
1513, 14eqeq12d 2297 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( ( card `  x )  =  x  <->  ( card `  ( aleph `  A ) )  =  ( aleph `  A
) ) )
16 sseq2 3200 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( om  C_  x  <->  om  C_  ( aleph `  A ) ) )
17 eqeq1 2289 . . . . . . 7  |-  ( x  =  ( aleph `  A
)  ->  ( x  =  ( aleph `  y
)  <->  ( aleph `  A
)  =  ( aleph `  y ) ) )
1817notbid 285 . . . . . 6  |-  ( x  =  ( aleph `  A
)  ->  ( -.  x  =  ( aleph `  y )  <->  -.  ( aleph `  A )  =  ( aleph `  y )
) )
1918ralbidv 2563 . . . . 5  |-  ( x  =  ( aleph `  A
)  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  ( aleph `  A )  =  ( aleph `  y )
) )
2015, 16, 193anbi123d 1252 . . . 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 2914 . . 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 1136 . 2  |-  ( A  e.  On  ->  ( aleph `  A )  e. 
{ x  |  ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) ) } )
23 cardalephex 7717 . . . . . . . . . 10  |-  ( om  C_  z  ->  ( (
card `  z )  =  z  <->  E. y  e.  On  z  =  ( aleph `  y ) ) )
2423biimpac 472 . . . . . . . . 9  |-  ( ( ( card `  z
)  =  z  /\  om  C_  z )  ->  E. y  e.  On  z  =  (
aleph `  y ) )
25 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( aleph `  y
)  ->  ( z  e.  ( aleph `  A )  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
26 alephord2 7703 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  e.  A  <->  (
aleph `  y )  e.  ( aleph `  A )
) )
2726bicomd 192 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( aleph `  y
)  e.  ( aleph `  A )  <->  y  e.  A ) )
2825, 27sylan9bbr 681 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( z  e.  ( aleph `  A )  <->  y  e.  A ) )
2928biimpcd 215 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  y  e.  A
) )
30 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) )
3130a1i 10 . . . . . . . . . . . . . 14  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  z  =  (
aleph `  y ) ) )
3229, 31jcad 519 . . . . . . . . . . . . 13  |-  ( z  e.  ( aleph `  A
)  ->  ( (
( y  e.  On  /\  A  e.  On )  /\  z  =  (
aleph `  y ) )  ->  ( y  e.  A  /\  z  =  ( aleph `  y )
) ) )
3332exp4c 591 . . . . . . . . . . . 12  |-  ( z  e.  ( aleph `  A
)  ->  ( y  e.  On  ->  ( A  e.  On  ->  ( z  =  ( aleph `  y
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) ) ) )
3433com3r 73 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  (
y  e.  On  ->  ( z  =  ( aleph `  y )  ->  (
y  e.  A  /\  z  =  ( aleph `  y ) ) ) ) ) )
3534imp4b 573 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( y  e.  On  /\  z  =  ( aleph `  y )
)  ->  ( y  e.  A  /\  z  =  ( aleph `  y
) ) ) )
3635reximdv2 2652 . . . . . . . . 9  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( E. y  e.  On  z  =  (
aleph `  y )  ->  E. y  e.  A  z  =  ( aleph `  y ) ) )
3724, 36syl5 28 . . . . . . . 8  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  -> 
( ( ( card `  z )  =  z  /\  om  C_  z
)  ->  E. y  e.  A  z  =  ( aleph `  y )
) )
3837imp 418 . . . . . . 7  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  E. y  e.  A  z  =  ( aleph `  y ) )
39 dfrex2 2556 . . . . . . 7  |-  ( E. y  e.  A  z  =  ( aleph `  y
)  <->  -.  A. y  e.  A  -.  z  =  ( aleph `  y
) )
4038, 39sylib 188 . . . . . 6  |-  ( ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  /\  ( ( card `  z
)  =  z  /\  om  C_  z ) )  ->  -.  A. y  e.  A  -.  z  =  ( aleph `  y ) )
41 nan 563 . . . . . 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 200 . . . . 5  |-  ( ( A  e.  On  /\  z  e.  ( aleph `  A ) )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
4342ex 423 . . . 4  |-  ( A  e.  On  ->  (
z  e.  ( aleph `  A )  ->  -.  ( ( ( card `  z )  =  z  /\  om  C_  z
)  /\  A. y  e.  A  -.  z  =  ( aleph `  y
) ) ) )
44 vex 2791 . . . . . . 7  |-  z  e. 
_V
45 fveq2 5525 . . . . . . . . 9  |-  ( x  =  z  ->  ( card `  x )  =  ( card `  z
) )
46 id 19 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
4745, 46eqeq12d 2297 . . . . . . . 8  |-  ( x  =  z  ->  (
( card `  x )  =  x  <->  ( card `  z
)  =  z ) )
48 sseq2 3200 . . . . . . . 8  |-  ( x  =  z  ->  ( om  C_  x  <->  om  C_  z
) )
49 eqeq1 2289 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  =  ( aleph `  y )  <->  z  =  ( aleph `  y )
) )
5049notbid 285 . . . . . . . . 9  |-  ( x  =  z  ->  ( -.  x  =  ( aleph `  y )  <->  -.  z  =  ( aleph `  y
) ) )
5150ralbidv 2563 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  A  -.  x  =  ( aleph `  y )  <->  A. y  e.  A  -.  z  =  ( aleph `  y
) ) )
5247, 48, 513anbi123d 1252 . . . . . . 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 2914 . . . . . 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 936 . . . . . 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 240 . . . . 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 287 . . . 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 218 . . 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 2625 . 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 7577 . . . . . 6  |-  ( card `  x )  e.  On
60 eleq1 2343 . . . . . 6  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
6159, 60mpbii 202 . . . . 5  |-  ( (
card `  x )  =  x  ->  x  e.  On )
62613ad2ant1 976 . . . 4  |-  ( ( ( card `  x
)  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y
) )  ->  x  e.  On )
6362abssi 3248 . . 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 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 642 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   |^|cint 3862   Oncon0 4392   omcom 4656   ` cfv 5255   cardccrd 7568   alephcale 7569
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-reg 7306  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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