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Theorem oncardval 6424
Description: The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
oncardval  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
Distinct variable group:    x, A

Proof of Theorem oncardval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 enrefg 6275 . . 3  |-  ( A  e.  On  ->  A  ~~  A )
2 breq1 3795 . . . 4  |-  ( y  =  A  ->  (
y  ~~  A  <->  A  ~~  A ) )
32rspcev 2673 . . 3  |-  ( ( A  e.  On  /\  A  ~~  A )  ->  E. y  e.  On  y  ~~  A )
41, 3mpdan 406 . 2  |-  ( A  e.  On  ->  E. y  e.  On  y  ~~  A
)
5 cardval3ex 6423 . 2  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
64, 5syl 14 1  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   E.wrex 2324   {crab 2327   |^|cint 3643   class class class wbr 3792   Oncon0 4128   ` cfv 4930    ~~ cen 6250   cardccrd 6417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-en 6253  df-card 6418
This theorem is referenced by:  cardonle  6425
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