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Definition df-har 8005
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 is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written ( ) and the cardinal successor ; 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 8342.

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
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-har
StepHypRef Expression
1 char 8003 . 2
2 vx . . 3
3 cvv 3109 . . 3
4 vy . . . . . 6
54cv 1394 . . . . 5
62cv 1394 . . . . 5
7 cdom 7534 . . . . 5
85, 6, 7wbr 4452 . . . 4
9 con0 4883 . . . 4
108, 4, 9crab 2811 . . 3
112, 3, 10cmpt 4510 . 2
121, 11wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  harf  8007  harval  8009
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