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Definition df-har 7984
Description: Define the Hartogs function , which maps all sets to the smallest ordinal that cannot be injected into the given set. In the important special case where  x is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written  aleph ( X ) and the cardinal successor 
X  +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 8321.

Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion
Ref Expression
df-har  |- har  =  ( x  e.  _V  |->  { y  e.  On  | 
y  ~<_  x } )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-har
StepHypRef Expression
1 char 7982 . 2  class har
2 vx . . 3  setvar  x
3 cvv 3113 . . 3  class  _V
4 vy . . . . . 6  setvar  y
54cv 1378 . . . . 5  class  y
62cv 1378 . . . . 5  class  x
7 cdom 7514 . . . . 5  class  ~<_
85, 6, 7wbr 4447 . . . 4  wff  y  ~<_  x
9 con0 4878 . . . 4  class  On
108, 4, 9crab 2818 . . 3  class  { y  e.  On  |  y  ~<_  x }
112, 3, 10cmpt 4505 . 2  class  ( x  e.  _V  |->  { y  e.  On  |  y  ~<_  x } )
121, 11wceq 1379 1  wff har  =  ( x  e.  _V  |->  { y  e.  On  | 
y  ~<_  x } )
Colors of variables: wff setvar class
This definition is referenced by:  harf  7986  harval  7988
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