Theorem prid1g

Description: An unordered pair contains its first member. Part of Theorem 7.6 of Quine p. 49. (Contributed by Stefan Allan, 8-Nov-2008)

	Ref	Expression
Assertion	prid1g	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 , 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2	1	orci	⊢ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 )
3		elprg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝐴 , 𝐵 } ↔ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) ) )
4	2 3	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 , 𝐵 } )