Metamath Proof Explorer


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 ( ( 𝐴𝑉𝐵𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )

Proof

Step Hyp Ref Expression
1 cnvcnvsn { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐴 , 𝐵 ⟩ }
2 relsnopg ( ( 𝐵𝑊𝐴𝑉 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } )
3 2 ancoms ( ( 𝐴𝑉𝐵𝑊 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } )
4 dfrel2 ( Rel { ⟨ 𝐵 , 𝐴 ⟩ } ↔ { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )
5 3 4 sylib ( ( 𝐴𝑉𝐵𝑊 ) → { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )
6 1 5 eqtr3id ( ( 𝐴𝑉𝐵𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } )