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Theorem cardval2 7883
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 8426. This theorem could be used to give us a simpler definition of  card in place of df-card 7831. 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 7866 . . . . . 6  |-  ( ( x  e.  On  /\  A  e.  dom  card )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
21ancoms 441 . . . . 5  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( x  ~<  A  <-> 
x  e.  ( card `  A ) ) )
32pm5.32da 624 . . . 4  |-  ( A  e.  dom  card  ->  ( ( x  e.  On  /\  x  ~<  A )  <->  ( x  e.  On  /\  x  e.  ( card `  A ) ) ) )
4 cardon 7836 . . . . . 6  |-  ( card `  A )  e.  On
54oneli 4692 . . . . 5  |-  ( x  e.  ( card `  A
)  ->  x  e.  On )
65pm4.71ri 616 . . . 4  |-  ( x  e.  ( card `  A
)  <->  ( x  e.  On  /\  x  e.  ( card `  A
) ) )
73, 6syl6rbbr 257 . . 3  |-  ( A  e.  dom  card  ->  ( x  e.  ( card `  A )  <->  ( x  e.  On  /\  x  ~<  A ) ) )
87abbi2dv 2553 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  |  ( x  e.  On  /\  x  ~<  A ) } )
9 df-rab 2716 . 2  |-  { x  e.  On  |  x  ~<  A }  =  { x  |  ( x  e.  On  /\  x  ~<  A ) }
108, 9syl6eqr 2488 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  e.  On  |  x  ~<  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711   class class class wbr 4215   Oncon0 4584   dom cdm 4881   ` cfv 5457    ~< csdm 7111   cardccrd 7827
This theorem is referenced by:  ondomon  8443  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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-rab 2716  df-v 2960  df-sbc 3164  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-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-we 4546  df-ord 4587  df-on 4588  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-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-card 7831
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