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Theorem cardval2 8384
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 8933. This theorem could be used to give us a simpler definition of  card in place of df-card 8332. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  e.  On  |  x  ~<  A }
)
Distinct variable group:    x, A

Proof of Theorem cardval2
StepHypRef Expression
1 cardsdomel 8367 . . . . . 6  |-  ( ( x  e.  On  /\  A  e.  dom  card )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
21ancoms 453 . . . . 5  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( x  ~<  A  <-> 
x  e.  ( card `  A ) ) )
32pm5.32da 641 . . . 4  |-  ( A  e.  dom  card  ->  ( ( x  e.  On  /\  x  ~<  A )  <->  ( x  e.  On  /\  x  e.  ( card `  A ) ) ) )
4 cardon 8337 . . . . . 6  |-  ( card `  A )  e.  On
54oneli 4991 . . . . 5  |-  ( x  e.  ( card `  A
)  ->  x  e.  On )
65pm4.71ri 633 . . . 4  |-  ( x  e.  ( card `  A
)  <->  ( x  e.  On  /\  x  e.  ( card `  A
) ) )
73, 6syl6rbbr 264 . . 3  |-  ( A  e.  dom  card  ->  ( x  e.  ( card `  A )  <->  ( x  e.  On  /\  x  ~<  A ) ) )
87abbi2dv 2604 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  |  ( x  e.  On  /\  x  ~<  A ) } )
9 df-rab 2826 . 2  |-  { x  e.  On  |  x  ~<  A }  =  { x  |  ( x  e.  On  /\  x  ~<  A ) }
108, 9syl6eqr 2526 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  e.  On  |  x  ~<  A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2821   class class class wbr 4453   Oncon0 4884   dom cdm 5005   ` cfv 5594    ~< csdm 7527   cardccrd 8328
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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-rab 2826  df-v 3120  df-sbc 3337  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-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-we 4846  df-ord 4887  df-on 4888  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-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-card 8332
This theorem is referenced by:  ondomon  8950  alephsuc3  8967
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