Metamath Proof Explorer


Theorem altopthg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012)

Ref Expression
Assertion altopthg ( ( 𝐴𝑉𝐵𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶𝐵 = 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 altopthsn ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) )
2 sneqbg ( 𝐴𝑉 → ( { 𝐴 } = { 𝐶 } ↔ 𝐴 = 𝐶 ) )
3 sneqbg ( 𝐵𝑊 → ( { 𝐵 } = { 𝐷 } ↔ 𝐵 = 𝐷 ) )
4 2 3 bi2anan9 ( ( 𝐴𝑉𝐵𝑊 ) → ( ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) ↔ ( 𝐴 = 𝐶𝐵 = 𝐷 ) ) )
5 1 4 syl5bb ( ( 𝐴𝑉𝐵𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶𝐵 = 𝐷 ) ) )