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Definition df-fin7 8973
Description: A set is VII-finite iff it cannot be infinitely well-ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
Assertion
Ref Expression
df-fin7 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fin7
StepHypRef Expression
1 cfin7 8966 . 2 class FinVII
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 vy . . . . . . 7 setvar 𝑦
54cv 1473 . . . . . 6 class 𝑦
6 cen 7815 . . . . . 6 class
73, 5, 6wbr 4577 . . . . 5 wff 𝑥𝑦
8 con0 5626 . . . . . 6 class On
9 com 6934 . . . . . 6 class ω
108, 9cdif 3536 . . . . 5 class (On ∖ ω)
117, 4, 10wrex 2896 . . . 4 wff 𝑦 ∈ (On ∖ ω)𝑥𝑦
1211wn 3 . . 3 wff ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦
1312, 2cab 2595 . 2 class {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
141, 13wceq 1474 1 wff FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  isfin7  8983
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