Theorem bnj1152

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	bnj1152	⊢ ( 𝑌 ∈ pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 𝑅 𝑋 ↔ 𝑌 𝑅 𝑋 ) )
2		df-bnj14	⊢ pred ( 𝑋 , 𝐴 , 𝑅 ) = { 𝑦 ∈ 𝐴 ∣ 𝑦 𝑅 𝑋 }
3	1 2	elrab2	⊢ ( 𝑌 ∈ pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) )