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Theorem alephsuc2 7854
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7406 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 7720. (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 7853 . . 3  |-  ( A  e.  On  ->  (
x  ~<_  ( aleph `  A
)  <->  x  ~<  ( aleph ` 
suc  A ) ) )
21rabbidv 2865 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~<_  (
aleph `  A ) }  =  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) } )
3 alephon 7843 . . . . . . 7  |-  ( aleph ` 
suc  A )  e.  On
43oneli 4603 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  e.  On )
5 alephcard 7844 . . . . . . . . 9  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
6 iscard 7755 . . . . . . . . 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 199 . . . . . . . 8  |-  ( (
aleph `  suc  A )  e.  On  /\  A. y  e.  ( aleph ` 
suc  A ) y 
~<  ( aleph `  suc  A ) )
87simpri 448 . . . . . . 7  |-  A. y  e.  ( aleph `  suc  A ) y  ~<  ( aleph ` 
suc  A )
98rspec 2692 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  ~<  (
aleph `  suc  A ) )
104, 9jca 518 . . . . 5  |-  ( y  e.  ( aleph `  suc  A )  ->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
11 sdomel 7151 . . . . . . 7  |-  ( ( y  e.  On  /\  ( aleph `  suc  A )  e.  On )  -> 
( y  ~<  ( aleph `  suc  A )  ->  y  e.  (
aleph `  suc  A ) ) )
123, 11mpan2 652 . . . . . 6  |-  ( y  e.  On  ->  (
y  ~<  ( aleph `  suc  A )  ->  y  e.  ( aleph `  suc  A ) ) )
1312imp 418 . . . . 5  |-  ( ( y  e.  On  /\  y  ~<  ( aleph `  suc  A ) )  ->  y  e.  ( aleph `  suc  A ) )
1410, 13impbii 180 . . . 4  |-  ( y  e.  ( aleph `  suc  A )  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
15 breq1 4128 . . . . 5  |-  ( x  =  y  ->  (
x  ~<  ( aleph `  suc  A )  <->  y  ~<  ( aleph `  suc  A ) ) )
1615elrab 3009 . . . 4  |-  ( y  e.  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) }  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
1714, 16bitr4i 243 . . 3  |-  ( y  e.  ( aleph `  suc  A )  <->  y  e.  {
x  e.  On  |  x  ~<  ( aleph `  suc  A ) } )
1817eqriv 2363 . 2  |-  ( aleph ` 
suc  A )  =  { x  e.  On  |  x  ~<  ( aleph ` 
suc  A ) }
192, 18syl6reqr 2417 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   {crab 2632   class class class wbr 4125   Oncon0 4495   suc csuc 4497   ` cfv 5358    ~<_ cdom 7004    ~< csdm 7005   cardccrd 7715   alephcale 7716
This theorem is referenced by:  alephsuc3  8349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-oi 7372  df-har 7419  df-card 7719  df-aleph 7720
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