Theorem pr2el1

Description: If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023)

	Ref	Expression
Assertion	pr2el1	⊢ ( { 𝐴 , 𝐵 } ≈ 2o → 𝐴 ∈ { 𝐴 , 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		pr2cv	⊢ ( { 𝐴 , 𝐵 } ≈ 2o → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
2	1	simpld	⊢ ( { 𝐴 , 𝐵 } ≈ 2o → 𝐴 ∈ V )
3		prid1g	⊢ ( 𝐴 ∈ V → 𝐴 ∈ { 𝐴 , 𝐵 } )
4	2 3	syl	⊢ ( { 𝐴 , 𝐵 } ≈ 2o → 𝐴 ∈ { 𝐴 , 𝐵 } )