Theorem cnvsng

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015) (Proof shortened by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	cnvsng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		cnvcnvsn	⊢ ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = ◡ { ⟨ 𝐴 , 𝐵 ⟩ }
2		relsnopg	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } )
3	2	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } )
4		dfrel2	⊢ ( Rel { ⟨ 𝐵 , 𝐴 ⟩ } ↔ ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )
5	3 4	sylib	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )
6	1 5	eqtr3id	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )