Theorem opideq

Description: Equality conditions for ordered pairs <. A , A >. and <. B , B >. . (Contributed by Peter Mazsa, 22-Jul-2019) (Revised by Thierry Arnoux, 16-Feb-2022)

	Ref	Expression
Assertion	opideq	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐴 ⟩ = ⟨ 𝐵 , 𝐵 ⟩ ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opthg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ( ⟨ 𝐴 , 𝐴 ⟩ = ⟨ 𝐵 , 𝐵 ⟩ ↔ ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐵 ) ) )
2	1	anidms	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐴 ⟩ = ⟨ 𝐵 , 𝐵 ⟩ ↔ ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐵 ) ) )
3		anidm	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐵 ) ↔ 𝐴 = 𝐵 )
4	2 3	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐴 ⟩ = ⟨ 𝐵 , 𝐵 ⟩ ↔ 𝐴 = 𝐵 ) )