Definition df-bnj19

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

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cB	⊢ 𝐵
1		cA	⊢ 𝐴
2		cR	⊢ 𝑅
3	1 0 2	w-bnj19	⊢ TrFo ( 𝐵 , 𝐴 , 𝑅 )
4		vx	⊢ 𝑥
5	4	cv	⊢ 𝑥
6	1 2 5	c-bnj14	⊢ pred ( 𝑥 , 𝐴 , 𝑅 )
7	6 0	wss	⊢ pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐵
8	7 4 0	wral	⊢ ∀ 𝑥 ∈ 𝐵 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐵
9	3 8	wb	⊢ ( TrFo ( 𝐵 , 𝐴 , 𝑅 ) ↔ ∀ 𝑥 ∈ 𝐵 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐵 )