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Definition df-har 8619
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 ℵ(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 8966.

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 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-har
StepHypRef Expression
1 char 8617 . 2 class har
2 vx . . 3 setvar 𝑥
3 cvv 3351 . . 3 class V
4 vy . . . . . 6 setvar 𝑦
54cv 1630 . . . . 5 class 𝑦
62cv 1630 . . . . 5 class 𝑥
7 cdom 8107 . . . . 5 class
85, 6, 7wbr 4786 . . . 4 wff 𝑦𝑥
9 con0 5866 . . . 4 class On
108, 4, 9crab 3065 . . 3 class {𝑦 ∈ On ∣ 𝑦𝑥}
112, 3, 10cmpt 4863 . 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
121, 11wceq 1631 1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
Colors of variables: wff setvar class
This definition is referenced by:  harf  8621  harval  8623
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