Theorem br1steq

Description: Uniqueness condition for the binary relation 1st . (Contributed by Scott Fenton, 11-Apr-2014) (Proof shortened by Mario Carneiro, 3-May-2015)

	Ref	Expression
Hypotheses	br1steq.1	⊢ 𝐴 ∈ V
	br1steq.2	⊢ 𝐵 ∈ V
Assertion	br1steq	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ 1st 𝐶 ↔ 𝐶 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		br1steq.1	⊢ 𝐴 ∈ V
2		br1steq.2	⊢ 𝐵 ∈ V
3		br1steqg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ 1st 𝐶 ↔ 𝐶 = 𝐴 ) )
4	1 2 3	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ 1st 𝐶 ↔ 𝐶 = 𝐴 )