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Theorem alephsuc2 7966
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7516 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 7832. (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 7965 . . 3  |-  ( A  e.  On  ->  (
x  ~<_  ( aleph `  A
)  <->  x  ~<  ( aleph ` 
suc  A ) ) )
21rabbidv 2950 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~<_  (
aleph `  A ) }  =  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) } )
3 alephon 7955 . . . . . . 7  |-  ( aleph ` 
suc  A )  e.  On
43oneli 4692 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  e.  On )
5 alephcard 7956 . . . . . . . . 9  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
6 iscard 7867 . . . . . . . . 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 201 . . . . . . . 8  |-  ( (
aleph `  suc  A )  e.  On  /\  A. y  e.  ( aleph ` 
suc  A ) y 
~<  ( aleph `  suc  A ) )
87simpri 450 . . . . . . 7  |-  A. y  e.  ( aleph `  suc  A ) y  ~<  ( aleph ` 
suc  A )
98rspec 2772 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  ~<  (
aleph `  suc  A ) )
104, 9jca 520 . . . . 5  |-  ( y  e.  ( aleph `  suc  A )  ->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
11 sdomel 7257 . . . . . . 7  |-  ( ( y  e.  On  /\  ( aleph `  suc  A )  e.  On )  -> 
( y  ~<  ( aleph `  suc  A )  ->  y  e.  (
aleph `  suc  A ) ) )
123, 11mpan2 654 . . . . . 6  |-  ( y  e.  On  ->  (
y  ~<  ( aleph `  suc  A )  ->  y  e.  ( aleph `  suc  A ) ) )
1312imp 420 . . . . 5  |-  ( ( y  e.  On  /\  y  ~<  ( aleph `  suc  A ) )  ->  y  e.  ( aleph `  suc  A ) )
1410, 13impbii 182 . . . 4  |-  ( y  e.  ( aleph `  suc  A )  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
15 breq1 4218 . . . . 5  |-  ( x  =  y  ->  (
x  ~<  ( aleph `  suc  A )  <->  y  ~<  ( aleph `  suc  A ) ) )
1615elrab 3094 . . . 4  |-  ( y  e.  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) }  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
1714, 16bitr4i 245 . . 3  |-  ( y  e.  ( aleph `  suc  A )  <->  y  e.  {
x  e.  On  |  x  ~<  ( aleph `  suc  A ) } )
1817eqriv 2435 . 2  |-  ( aleph ` 
suc  A )  =  { x  e.  On  |  x  ~<  ( aleph ` 
suc  A ) }
192, 18syl6reqr 2489 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   class class class wbr 4215   Oncon0 4584   suc csuc 4586   ` cfv 5457    ~<_ cdom 7110    ~< csdm 7111   cardccrd 7827   alephcale 7828
This theorem is referenced by:  alephsuc3  8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-har 7529  df-card 7831  df-aleph 7832
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