Theorem opth1g

Description: Equality of the first members of equal ordered pairs. Closed form of opth1 . (Contributed by AV, 14-Oct-2018)

	Ref	Expression
Assertion	opth1g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ → 𝐴 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		opthg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
2		simpl	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → 𝐴 = 𝐶 )
3	1 2	syl6bi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ → 𝐴 = 𝐶 ) )