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Definition df-har 7288
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 7589.

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 7286 . 2  class har
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
4 vy . . . . . 6  set  y
54cv 1631 . . . . 5  class  y
62cv 1631 . . . . 5  class  x
7 cdom 6877 . . . . 5  class  ~<_
85, 6, 7wbr 4039 . . . 4  wff  y  ~<_  x
9 con0 4408 . . . 4  class  On
108, 4, 9crab 2560 . . 3  class  { y  e.  On  |  y  ~<_  x }
112, 3, 10cmpt 4093 . 2  class  ( x  e.  _V  |->  { y  e.  On  |  y  ~<_  x } )
121, 11wceq 1632 1  wff har  =  ( x  e.  _V  |->  { y  e.  On  | 
y  ~<_  x } )
Colors of variables: wff set class
This definition is referenced by:  harf  7290  harval  7292
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