Definition df-har 9025

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 9372.

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 9023	. 2 class har
2		vx	. . 3 setvar 𝑥
3		cvv 3497	. . 3 class V
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1535	. . . . 5 class 𝑦
6	2	cv 1535	. . . . 5 class 𝑥
7		cdom 8510	. . . . 5 class ≼
8	5, 6, 7	wbr 5069	. . . 4 wff 𝑦 ≼ 𝑥
9		con0 6194	. . . 4 class On
10	8, 4, 9	crab 3145	. . 3 class {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}
11	2, 3, 10	cmpt 5149	. 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
12	1, 11	wceq 1536	1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})

This definition is referenced by:  harf  9027  harval  9029