Definition df-fin 6637

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 13174. (Contributed by NM, 22-Aug-2008.)

Assertion

Ref	Expression
df-fin	⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fin

Step	Hyp	Ref	Expression
1		cfn 6634	. 2 class Fin
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1330	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1330	. . . . 5 class 𝑦
6		cen 6632	. . . . 5 class ≈
7	3, 5, 6	wbr 3929	. . . 4 wff 𝑥 ≈ 𝑦
8		com 4504	. . . 4 class ω
9	7, 4, 8	wrex 2417	. . 3 wff ∃𝑦 ∈ ω 𝑥 ≈ 𝑦
10	9, 2	cab 2125	. 2 class {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
11	1, 10	wceq 1331	1 wff Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}

This definition is referenced by:  isfi  6655