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Theorem alephsuc2 7921
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7473 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 7787. (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 7920 . . 3  |-  ( A  e.  On  ->  (
x  ~<_  ( aleph `  A
)  <->  x  ~<  ( aleph ` 
suc  A ) ) )
21rabbidv 2912 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~<_  (
aleph `  A ) }  =  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) } )
3 alephon 7910 . . . . . . 7  |-  ( aleph ` 
suc  A )  e.  On
43oneli 4652 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  e.  On )
5 alephcard 7911 . . . . . . . . 9  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
6 iscard 7822 . . . . . . . . 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 200 . . . . . . . 8  |-  ( (
aleph `  suc  A )  e.  On  /\  A. y  e.  ( aleph ` 
suc  A ) y 
~<  ( aleph `  suc  A ) )
87simpri 449 . . . . . . 7  |-  A. y  e.  ( aleph `  suc  A ) y  ~<  ( aleph ` 
suc  A )
98rspec 2734 . . . . . 6  |-  ( y  e.  ( aleph `  suc  A )  ->  y  ~<  (
aleph `  suc  A ) )
104, 9jca 519 . . . . 5  |-  ( y  e.  ( aleph `  suc  A )  ->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
11 sdomel 7217 . . . . . . 7  |-  ( ( y  e.  On  /\  ( aleph `  suc  A )  e.  On )  -> 
( y  ~<  ( aleph `  suc  A )  ->  y  e.  (
aleph `  suc  A ) ) )
123, 11mpan2 653 . . . . . 6  |-  ( y  e.  On  ->  (
y  ~<  ( aleph `  suc  A )  ->  y  e.  ( aleph `  suc  A ) ) )
1312imp 419 . . . . 5  |-  ( ( y  e.  On  /\  y  ~<  ( aleph `  suc  A ) )  ->  y  e.  ( aleph `  suc  A ) )
1410, 13impbii 181 . . . 4  |-  ( y  e.  ( aleph `  suc  A )  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
15 breq1 4179 . . . . 5  |-  ( x  =  y  ->  (
x  ~<  ( aleph `  suc  A )  <->  y  ~<  ( aleph `  suc  A ) ) )
1615elrab 3056 . . . 4  |-  ( y  e.  { x  e.  On  |  x  ~<  (
aleph `  suc  A ) }  <->  ( y  e.  On  /\  y  ~< 
( aleph `  suc  A ) ) )
1714, 16bitr4i 244 . . 3  |-  ( y  e.  ( aleph `  suc  A )  <->  y  e.  {
x  e.  On  |  x  ~<  ( aleph `  suc  A ) } )
1817eqriv 2405 . 2  |-  ( aleph ` 
suc  A )  =  { x  e.  On  |  x  ~<  ( aleph ` 
suc  A ) }
192, 18syl6reqr 2459 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   {crab 2674   class class class wbr 4176   Oncon0 4545   suc csuc 4547   ` cfv 5417    ~<_ cdom 7070    ~< csdm 7071   cardccrd 7782   alephcale 7783
This theorem is referenced by:  alephsuc3  8415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-card 7786  df-aleph 7787
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