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Definition df-fin 7822
Description: Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our "𝑎 ∈ Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 8398. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (theorem isfinite 8409.) (Contributed by NM, 22-Aug-2008.)
Assertion
Ref Expression
df-fin Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fin
StepHypRef Expression
1 cfn 7818 . 2 class Fin
2 vx . . . . . 6 setvar 𝑥
32cv 1473 . . . . 5 class 𝑥
4 vy . . . . . 6 setvar 𝑦
54cv 1473 . . . . 5 class 𝑦
6 cen 7815 . . . . 5 class
73, 5, 6wbr 4577 . . . 4 wff 𝑥𝑦
8 com 6934 . . . 4 class ω
97, 4, 8wrex 2896 . . 3 wff 𝑦 ∈ ω 𝑥𝑦
109, 2cab 2595 . 2 class {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
111, 10wceq 1474 1 wff Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  isfi  7842  dffin1-5  9070
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