Definition df-bnj19 32576

Description: Define the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
df-bnj19	⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴   𝑥,𝑅

Detailed syntax breakdown of Definition df-bnj19

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		cR	. . 3 class 𝑅
4	1, 2, 3	w-bnj19 32575	. 2 wff TrFo(𝐵, 𝐴, 𝑅)
5		vx	. . . . . 6 setvar 𝑥
6	5	cv 1538	. . . . 5 class 𝑥
7	1, 3, 6	c-bnj14 32567	. . . 4 class pred(𝑥, 𝐴, 𝑅)
8	7, 2	wss 3883	. . 3 wff pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵
9	8, 5, 2	wral 3063	. 2 wff ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵
10	4, 9	wb 205	1 wff ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)

This definition is referenced by:  bnj978  32829  bnj1118  32864  bnj1125  32872  bnj1137  32875  bnj1408  32916