Metamath Proof Explorer


Theorem brtpid3

Description: A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016)

Ref Expression
Assertion brtpid3 𝐴 { 𝐶 , 𝐷 , ⟨ 𝐴 , 𝐵 ⟩ } 𝐵

Proof

Step Hyp Ref Expression
1 opex 𝐴 , 𝐵 ⟩ ∈ V
2 1 tpid3 𝐴 , 𝐵 ⟩ ∈ { 𝐶 , 𝐷 , ⟨ 𝐴 , 𝐵 ⟩ }
3 df-br ( 𝐴 { 𝐶 , 𝐷 , ⟨ 𝐴 , 𝐵 ⟩ } 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { 𝐶 , 𝐷 , ⟨ 𝐴 , 𝐵 ⟩ } )
4 2 3 mpbir 𝐴 { 𝐶 , 𝐷 , ⟨ 𝐴 , 𝐵 ⟩ } 𝐵