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Theorem cardval2 7711
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 8255. This theorem could be used to give us a simpler definition of  card in place of df-card 7659. 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 7694 . . . . . 6  |-  ( ( x  e.  On  /\  A  e.  dom  card )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
21ancoms 439 . . . . 5  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( x  ~<  A  <-> 
x  e.  ( card `  A ) ) )
32pm5.32da 622 . . . 4  |-  ( A  e.  dom  card  ->  ( ( x  e.  On  /\  x  ~<  A )  <->  ( x  e.  On  /\  x  e.  ( card `  A ) ) ) )
4 cardon 7664 . . . . . 6  |-  ( card `  A )  e.  On
54oneli 4579 . . . . 5  |-  ( x  e.  ( card `  A
)  ->  x  e.  On )
65pm4.71ri 614 . . . 4  |-  ( x  e.  ( card `  A
)  <->  ( x  e.  On  /\  x  e.  ( card `  A
) ) )
73, 6syl6rbbr 255 . . 3  |-  ( A  e.  dom  card  ->  ( x  e.  ( card `  A )  <->  ( x  e.  On  /\  x  ~<  A ) ) )
87abbi2dv 2473 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  |  ( x  e.  On  /\  x  ~<  A ) } )
9 df-rab 2628 . 2  |-  { x  e.  On  |  x  ~<  A }  =  { x  |  ( x  e.  On  /\  x  ~<  A ) }
108, 9syl6eqr 2408 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  e.  On  |  x  ~<  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   {crab 2623   class class class wbr 4102   Oncon0 4471   dom cdm 4768   ` cfv 5334    ~< csdm 6947   cardccrd 7655
This theorem is referenced by:  ondomon  8272  alephsuc3  8289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-card 7659
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