MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-card Structured version   Visualization version   GIF version

Definition df-card 9357
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 9957 for its value and cardval2 9409 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 9962. 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 9353 . 2 class card
2 vx . . 3 setvar 𝑥
3 cvv 3495 . . 3 class V
4 vy . . . . . . 7 setvar 𝑦
54cv 1527 . . . . . 6 class 𝑦
62cv 1527 . . . . . 6 class 𝑥
7 cen 8495 . . . . . 6 class
85, 6, 7wbr 5058 . . . . 5 wff 𝑦𝑥
9 con0 6185 . . . . 5 class On
108, 4, 9crab 3142 . . . 4 class {𝑦 ∈ On ∣ 𝑦𝑥}
1110cint 4869 . . 3 class {𝑦 ∈ On ∣ 𝑦𝑥}
122, 3, 11cmpt 5138 . 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
131, 12wceq 1528 1 wff card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
Colors of variables: wff setvar class
This definition is referenced by:  cardf2  9361  cardval3  9370  iscard4  39780  harval3  39784
  Copyright terms: Public domain W3C validator