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Theorem alephsuc2 8473
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7981 function by transfinite recursion, starting from 
om. Using this theorem we could define the aleph function with  { z  e.  On  |  z  ~<_  x } in place of  |^| { z  e.  On  |  x 
~<  z } in df-aleph 8333. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephsuc2  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
Distinct variable group:    x, A

Proof of Theorem alephsuc2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 alephsucdom 8472 . . 3  |-  ( A  e.  On  ->  (
x  ~<_  ( aleph `  A
)  <->  x  ~<  ( aleph ` 
suc  A ) ) )
21rabbidv 3110 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~<_  (
aleph `  A ) }  =  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) } )
3 alephon 8462 . . . . . . 7  |-  ( aleph ` 
suc  A )  e.  On
43oneli 4991 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  e.  On )
5 alephcard 8463 . . . . . . . . 9  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
6 iscard 8368 . . . . . . . . 9  |-  ( (
card `  ( aleph `  suc  A ) )  =  (
aleph `  suc  A )  <-> 
( ( aleph `  suc  A )  e.  On  /\  A. y  e.  ( aleph ` 
suc  A ) y 
~<  ( aleph `  suc  A ) ) )
75, 6mpbi 208 . . . . . . . 8  |-  ( (
aleph `  suc  A )  e.  On  /\  A. y  e.  ( aleph ` 
suc  A ) y 
~<  ( aleph `  suc  A ) )
87simpri 462 . . . . . . 7  |-  A. y  e.  ( aleph `  suc  A ) y  ~<  ( aleph ` 
suc  A )
98rspec 2835 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  ~<  (
aleph `  suc  A ) )
104, 9jca 532 . . . . 5  |-  ( y  e.  ( aleph `  suc  A )  ->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
11 sdomel 7676 . . . . . . 7  |-  ( ( y  e.  On  /\  ( aleph `  suc  A )  e.  On )  -> 
( y  ~<  ( aleph `  suc  A )  ->  y  e.  (
aleph `  suc  A ) ) )
123, 11mpan2 671 . . . . . 6  |-  ( y  e.  On  ->  (
y  ~<  ( aleph `  suc  A )  ->  y  e.  ( aleph `  suc  A ) ) )
1312imp 429 . . . . 5  |-  ( ( y  e.  On  /\  y  ~<  ( aleph `  suc  A ) )  ->  y  e.  ( aleph `  suc  A ) )
1410, 13impbii 188 . . . 4  |-  ( y  e.  ( aleph `  suc  A )  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
15 breq1 4456 . . . . 5  |-  ( x  =  y  ->  (
x  ~<  ( aleph `  suc  A )  <->  y  ~<  ( aleph `  suc  A ) ) )
1615elrab 3266 . . . 4  |-  ( y  e.  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) }  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
1714, 16bitr4i 252 . . 3  |-  ( y  e.  ( aleph `  suc  A )  <->  y  e.  {
x  e.  On  |  x  ~<  ( aleph `  suc  A ) } )
1817eqriv 2463 . 2  |-  ( aleph ` 
suc  A )  =  { x  e.  On  |  x  ~<  ( aleph ` 
suc  A ) }
192, 18syl6reqr 2527 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   class class class wbr 4453   Oncon0 4884   suc csuc 4886   ` cfv 5594    ~<_ cdom 7526    ~< csdm 7527   cardccrd 8328   alephcale 8329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-har 7996  df-card 8332  df-aleph 8333
This theorem is referenced by:  alephsuc3  8967
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