Theorem elons

Description: Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025)

	Ref	Expression
Assertion	elons	⊢ ( 𝐴 ∈ Ons ↔ ( 𝐴 ∈ No ∧ ( R ‘ 𝐴 ) = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑥 = 𝐴 → ( R ‘ 𝑥 ) = ( R ‘ 𝐴 ) )
2	1	eqeq1d	⊢ ( 𝑥 = 𝐴 → ( ( R ‘ 𝑥 ) = ∅ ↔ ( R ‘ 𝐴 ) = ∅ ) )
3		df-ons	⊢ Ons = { 𝑥 ∈ No ∣ ( R ‘ 𝑥 ) = ∅ }
4	2 3	elrab2	⊢ ( 𝐴 ∈ Ons ↔ ( 𝐴 ∈ No ∧ ( R ‘ 𝐴 ) = ∅ ) )