Theorem opidg

Description: The ordered pair <. A , A >. in Kuratowski's representation. Closed form of opid . (Contributed by Peter Mazsa, 22-Jul-2019) (Avoid depending on this detail.)

	Ref	Expression
Assertion	opidg	⊢ ( 𝐴 ∈ 𝑉 → ⟨ 𝐴 , 𝐴 ⟩ = { { 𝐴 } } )

Proof

Step	Hyp	Ref	Expression
1		dfopg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ⟨ 𝐴 , 𝐴 ⟩ = { { 𝐴 } , { 𝐴 , 𝐴 } } )
2	1	anidms	⊢ ( 𝐴 ∈ 𝑉 → ⟨ 𝐴 , 𝐴 ⟩ = { { 𝐴 } , { 𝐴 , 𝐴 } } )
3		dfsn2	⊢ { 𝐴 } = { 𝐴 , 𝐴 }
4	3	eqcomi	⊢ { 𝐴 , 𝐴 } = { 𝐴 }
5	4	preq2i	⊢ { { 𝐴 } , { 𝐴 , 𝐴 } } = { { 𝐴 } , { 𝐴 } }
6		dfsn2	⊢ { { 𝐴 } } = { { 𝐴 } , { 𝐴 } }
7	5 6	eqtr4i	⊢ { { 𝐴 } , { 𝐴 , 𝐴 } } = { { 𝐴 } }
8	2 7	eqtrdi	⊢ ( 𝐴 ∈ 𝑉 → ⟨ 𝐴 , 𝐴 ⟩ = { { 𝐴 } } )