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Theorem oncardval 6814
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 6481 . . 3  |-  ( A  e.  On  ->  A  ~~  A )
2 breq1 3848 . . . 4  |-  ( y  =  A  ->  (
y  ~~  A  <->  A  ~~  A ) )
32rspcev 2722 . . 3  |-  ( ( A  e.  On  /\  A  ~~  A )  ->  E. y  e.  On  y  ~~  A )
41, 3mpdan 412 . 2  |-  ( A  e.  On  ->  E. y  e.  On  y  ~~  A
)
5 cardval3ex 6813 . 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 1289    e. wcel 1438   E.wrex 2360   {crab 2363   |^|cint 3688   class class class wbr 3845   Oncon0 4190   ` cfv 5015    ~~ cen 6455   cardccrd 6807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-en 6458  df-card 6808
This theorem is referenced by:  cardonle  6815
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