Theorem elright

Description: Membership in the right set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	elright	⊢ ( 𝐴 ∈ ( R ‘ 𝐵 ) ↔ ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∧ 𝐵 <s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑥 = 𝐴 → ( 𝐵 <s 𝑥 ↔ 𝐵 <s 𝐴 ) )
2		rightval	⊢ ( R ‘ 𝐵 ) = { 𝑥 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∣ 𝐵 <s 𝑥 }
3	1 2	elrab2	⊢ ( 𝐴 ∈ ( R ‘ 𝐵 ) ↔ ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∧ 𝐵 <s 𝐴 ) )