Definition df-har 9005

Description: Define the Hartogs function as mapping a set to the class of ordinals it dominates. That class is an ordinal by hartogs 8992, which is used in harf 9006.

The Hartogs number of a set is the least ordinal not dominated by that set. Theorem harval2 9410 proves that the Hartogs function actually gives the Hartogs number for well-orderable sets.

The Hartogs number of an ordinal is its cardinal successor. This is proved for finite ordinal in harsucnn 9411.

Traditionally, the Hartogs number of a set 𝑋 is written ℵ(𝑋), and its 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 9353.

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 ∣ 𝑦 ≼ 𝑥})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-har

Step	Hyp	Ref	Expression
1		char 9004	. 2 class har
2		vx	. . 3 setvar 𝑥
3		cvv 3441	. . 3 class V
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1537	. . . . 5 class 𝑦
6	2	cv 1537	. . . . 5 class 𝑥
7		cdom 8490	. . . . 5 class ≼
8	5, 6, 7	wbr 5030	. . . 4 wff 𝑦 ≼ 𝑥
9		con0 6159	. . . 4 class On
10	8, 4, 9	crab 3110	. . 3 class {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}
11	2, 3, 10	cmpt 5110	. 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
12	1, 11	wceq 1538	1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})

This definition is referenced by:  harf  9006  harval  9008