Theorem eqrelriiv

Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995)

	Ref	Expression
Hypotheses	eqrelriiv.1	⊢ Rel 𝐴
	eqrelriiv.2	⊢ Rel 𝐵
	eqrelriiv.3	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
Assertion	eqrelriiv	⊢ 𝐴 = 𝐵

Proof

Step	Hyp	Ref	Expression
1		eqrelriiv.1	⊢ Rel 𝐴
2		eqrelriiv.2	⊢ Rel 𝐵
3		eqrelriiv.3	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
4	3	eqrelriv	⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → 𝐴 = 𝐵 )
5	1 2 4	mp2an	⊢ 𝐴 = 𝐵