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Mirrors > Home > MPE Home > Th. List > df-har | Structured version Visualization version GIF version |
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 8766. 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 |
---|---|
df-har | ⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | char 8461 | . 2 class har | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3200 | . . 3 class V | |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1482 | . . . . 5 class 𝑦 |
6 | 2 | cv 1482 | . . . . 5 class 𝑥 |
7 | cdom 7953 | . . . . 5 class ≼ | |
8 | 5, 6, 7 | wbr 4653 | . . . 4 wff 𝑦 ≼ 𝑥 |
9 | con0 5723 | . . . 4 class On | |
10 | 8, 4, 9 | crab 2916 | . . 3 class {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥} |
11 | 2, 3, 10 | cmpt 4729 | . 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
12 | 1, 11 | wceq 1483 | 1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
Colors of variables: wff setvar class |
This definition is referenced by: harf 8465 harval 8467 |
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