Definition df-har

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

TheHartogs number of a set is the least ordinal not dominated by that set. Theorem harval2 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 .

Traditionally, the Hartogs number of a set X is written aleph ( 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 .

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)

	Ref	Expression
Assertion	df-har	⊢ har = ( 𝑥 ∈ V ↦ { 𝑦 ∈ On ∣ 𝑦 ≼ 𝑥 } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		char	⊢ har
1		vx	⊢ 𝑥
2		cvv	⊢ V
3		vy	⊢ 𝑦
4		con0	⊢ On
5	3	cv	⊢ 𝑦
6		cdom	⊢ ≼
7	1	cv	⊢ 𝑥
8	5 7 6	wbr	⊢ 𝑦 ≼ 𝑥
9	8 3 4	crab	⊢ { 𝑦 ∈ On ∣ 𝑦 ≼ 𝑥 }
10	1 2 9	cmpt	⊢ ( 𝑥 ∈ V ↦ { 𝑦 ∈ On ∣ 𝑦 ≼ 𝑥 } )
11	0 10	wceq	⊢ har = ( 𝑥 ∈ V ↦ { 𝑦 ∈ On ∣ 𝑦 ≼ 𝑥 } )