Definition df-bj-ind 7150

Description: Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
df-bj-ind	⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A))

Distinct variable group:   x,A

Detailed syntax breakdown of Definition df-bj-ind

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2	1	wind 7149	. 2 wff Ind A
3		c0 3201	. . . 4 class ∅
4	3, 1	wcel 1374	. . 3 wff ∅ ∈ A
5		vx	. . . . . . 7 setvar x
6	5	cv 1227	. . . . . 6 class x
7	6	csuc 4051	. . . . 5 class suc x
8	7, 1	wcel 1374	. . . 4 wff suc x ∈ A
9	8, 5, 1	wral 2284	. . 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))

This definition is referenced by:  bj-indsuc  7151  bj-indeq  7152  bj-bdind  7153  bj-indint  7154  bj-dfom  7155  bj-inf2vnlem1  7188  bj-inf2vnlem2  7189