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Mirrors > Home > ILE Home > Th. List > Mathboxes > df-bj-ind | GIF version |
Description: Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
df-bj-ind | ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class A | |
2 | 1 | wind 9385 | . 2 wff Ind A |
3 | c0 3218 | . . . 4 class ∅ | |
4 | 3, 1 | wcel 1390 | . . 3 wff ∅ ∈ A |
5 | vx | . . . . . . 7 setvar x | |
6 | 5 | cv 1241 | . . . . . 6 class x |
7 | 6 | csuc 4068 | . . . . 5 class suc x |
8 | 7, 1 | wcel 1390 | . . . 4 wff suc x ∈ A |
9 | 8, 5, 1 | wral 2300 | . . 3 wff ∀x ∈ A suc x ∈ A |
10 | 4, 9 | wa 97 | . 2 wff (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) |
11 | 2, 10 | wb 98 | 1 wff (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) |
Colors of variables: wff set class |
This definition is referenced by: bj-indsuc 9387 bj-indeq 9388 bj-bdind 9389 bj-indint 9390 bj-indind 9391 bj-dfom 9392 peano5setOLD 9400 bj-inf2vnlem1 9430 bj-inf2vnlem2 9431 |
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