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Definition df-ihash 10033
Description: Define the set size function , which gives the cardinality of a finite set as a member of 0, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3.

Note that we use the sharp sign () for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 7977). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of and ) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

Assertion
Ref Expression
df-ihash ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ihash
StepHypRef Expression
1 chash 10032 . 2 class
2 vx . . . . . 6 setvar 𝑥
3 cz 8660 . . . . . 6 class
42cv 1286 . . . . . . 7 class 𝑥
5 c1 7272 . . . . . . 7 class 1
6 caddc 7274 . . . . . . 7 class +
74, 5, 6co 5594 . . . . . 6 class (𝑥 + 1)
82, 3, 7cmpt 3868 . . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9 cc0 7271 . . . . 5 class 0
108, 9cfrec 6090 . . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11 com 4371 . . . . . 6 class ω
12 cpnf 7440 . . . . . 6 class +∞
1311, 12cop 3428 . . . . 5 class ⟨ω, +∞⟩
1413csn 3425 . . . 4 class {⟨ω, +∞⟩}
1510, 14cun 2984 . . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16 cvv 2614 . . . 4 class V
17 vy . . . . . . . 8 setvar 𝑦
1817cv 1286 . . . . . . 7 class 𝑦
19 cdom 6389 . . . . . . 7 class
2018, 4, 19wbr 3814 . . . . . 6 wff 𝑦𝑥
2111csn 3425 . . . . . . 7 class {ω}
2211, 21cun 2984 . . . . . 6 class (ω ∪ {ω})
2320, 17, 22crab 2359 . . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}
2423cuni 3630 . . . 4 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}
252, 16, 24cmpt 3868 . . 3 class (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
2615, 25ccom 4408 . 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
271, 26wceq 1287 1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
Colors of variables: wff set class
This definition is referenced by:  hashinfom  10035  hashennn  10037
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