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Theorem alephsuc2 8364
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7872 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 8224. (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 8363 . . 3  |-  ( A  e.  On  ->  (
x  ~<_  ( aleph `  A
)  <->  x  ~<  ( aleph ` 
suc  A ) ) )
21rabbidv 3070 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~<_  (
aleph `  A ) }  =  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) } )
3 alephon 8353 . . . . . . 7  |-  ( aleph ` 
suc  A )  e.  On
43oneli 4937 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  e.  On )
5 alephcard 8354 . . . . . . . . 9  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
6 iscard 8259 . . . . . . . . 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 2898 . . . . . 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 7571 . . . . . . 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 4406 . . . . 5  |-  ( x  =  y  ->  (
x  ~<  ( aleph `  suc  A )  <->  y  ~<  ( aleph `  suc  A ) ) )
1615elrab 3224 . . . 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 2450 . 2  |-  ( aleph ` 
suc  A )  =  { x  e.  On  |  x  ~<  ( aleph ` 
suc  A ) }
192, 18syl6reqr 2514 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 1370    e. wcel 1758   A.wral 2799   {crab 2803   class class class wbr 4403   Oncon0 4830   suc csuc 4832   ` cfv 5529    ~<_ cdom 7421    ~< csdm 7422   cardccrd 8219   alephcale 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7838  df-har 7887  df-card 8223  df-aleph 8224
This theorem is referenced by:  alephsuc3  8858
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