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Definition df-card 9365
 Description: Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 9966 for its value and cardval2 9417 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 9971. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
df-card card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-card
StepHypRef Expression
1 ccrd 9361 . 2 class card
2 vx . . 3 setvar 𝑥
3 cvv 3480 . . 3 class V
4 vy . . . . . . 7 setvar 𝑦
54cv 1537 . . . . . 6 class 𝑦
62cv 1537 . . . . . 6 class 𝑥
7 cen 8502 . . . . . 6 class
85, 6, 7wbr 5052 . . . . 5 wff 𝑦𝑥
9 con0 6178 . . . . 5 class On
108, 4, 9crab 3137 . . . 4 class {𝑦 ∈ On ∣ 𝑦𝑥}
1110cint 4862 . . 3 class {𝑦 ∈ On ∣ 𝑦𝑥}
122, 3, 11cmpt 5132 . 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
131, 12wceq 1538 1 wff card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
 Colors of variables: wff setvar class This definition is referenced by:  cardf2  9369  cardval3  9378  iscard4  40157  harval3  40160
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