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Theorem cardval2 7624
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 8168. This theorem could be used to give us a simpler definition of  card in place of df-card 7572. 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 7607 . . . . . 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 7577 . . . . . 6  |-  ( card `  A )  e.  On
54oneli 4500 . . . . 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 2398 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  { x  |  ( x  e.  On  /\  x  ~<  A ) } )
9 df-rab 2552 . 2  |-  { x  e.  On  |  x  ~<  A }  =  { x  |  ( x  e.  On  /\  x  ~<  A ) }
108, 9syl6eqr 2333 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 1623    e. wcel 1684   {cab 2269   {crab 2547   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  ondomon  8185  alephsuc3  8202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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