Metamath Proof Explorer


Theorem altopth1

Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012)

Ref Expression
Assertion altopth1 ( 𝐴𝑉 → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ → 𝐴 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 altopthsn ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) )
2 sneqrg ( 𝐴𝑉 → ( { 𝐴 } = { 𝐶 } → 𝐴 = 𝐶 ) )
3 2 adantrd ( 𝐴𝑉 → ( ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) → 𝐴 = 𝐶 ) )
4 1 3 syl5bi ( 𝐴𝑉 → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ → 𝐴 = 𝐶 ) )