Definition df-har 9325

Description: Define the Hartogs function as mapping a set to the class of ordinals it dominates. That class is an ordinal by hartogs 9312, which is used in harf 9326.

The Hartogs number of a set is the least ordinal not dominated by that set. Theorem harval2 9764 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 9765.

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 9707.

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 9324	. 2 class har
2		vx	. . 3 setvar 𝑥
3		cvv 3433	. . 3 class V
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1538	. . . . 5 class 𝑦
6	2	cv 1538	. . . . 5 class 𝑥
7		cdom 8740	. . . . 5 class ≼
8	5, 6, 7	wbr 5075	. . . 4 wff 𝑦 ≼ 𝑥
9		con0 6270	. . . 4 class On
10	8, 4, 9	crab 3069	. . 3 class {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}
11	2, 3, 10	cmpt 5158	. 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
12	1, 11	wceq 1539	1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})

This definition is referenced by:  harf  9326  harval  9328