Metamath Proof Explorer


Theorem dfpr2

Description: Alternate definition of a pair. Definition 5.1 of TakeutiZaring p. 15. (Contributed by NM, 24-Apr-1994)

Ref Expression
Assertion dfpr2 { 𝐴 , 𝐵 } = { 𝑥 ∣ ( 𝑥 = 𝐴𝑥 = 𝐵 ) }

Proof

Step Hyp Ref Expression
1 df-pr { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
2 elun ( 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) ↔ ( 𝑥 ∈ { 𝐴 } ∨ 𝑥 ∈ { 𝐵 } ) )
3 velsn ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 )
4 velsn ( 𝑥 ∈ { 𝐵 } ↔ 𝑥 = 𝐵 )
5 3 4 orbi12i ( ( 𝑥 ∈ { 𝐴 } ∨ 𝑥 ∈ { 𝐵 } ) ↔ ( 𝑥 = 𝐴𝑥 = 𝐵 ) )
6 2 5 bitri ( 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) ↔ ( 𝑥 = 𝐴𝑥 = 𝐵 ) )
7 6 abbi2i ( { 𝐴 } ∪ { 𝐵 } ) = { 𝑥 ∣ ( 𝑥 = 𝐴𝑥 = 𝐵 ) }
8 1 7 eqtri { 𝐴 , 𝐵 } = { 𝑥 ∣ ( 𝑥 = 𝐴𝑥 = 𝐵 ) }