Theorem oncutleft

Description: A surreal ordinal is equal to the cut of its left set and the empty set. (Contributed by Scott Fenton, 29-Mar-2025)

	Ref	Expression
Assertion	oncutleft	⊢ ( 𝐴 ∈ Ons → 𝐴 = ( ( L ‘ 𝐴 ) |s ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		onno	⊢ ( 𝐴 ∈ Ons → 𝐴 ∈ No )
2		lrcut	⊢ ( 𝐴 ∈ No → ( ( L ‘ 𝐴 ) |s ( R ‘ 𝐴 ) ) = 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ Ons → ( ( L ‘ 𝐴 ) |s ( R ‘ 𝐴 ) ) = 𝐴 )
4		elons	⊢ ( 𝐴 ∈ Ons ↔ ( 𝐴 ∈ No ∧ ( R ‘ 𝐴 ) = ∅ ) )
5	4	simprbi	⊢ ( 𝐴 ∈ Ons → ( R ‘ 𝐴 ) = ∅ )
6	5	oveq2d	⊢ ( 𝐴 ∈ Ons → ( ( L ‘ 𝐴 ) |s ( R ‘ 𝐴 ) ) = ( ( L ‘ 𝐴 ) |s ∅ ) )
7	3 6	eqtr3d	⊢ ( 𝐴 ∈ Ons → 𝐴 = ( ( L ‘ 𝐴 ) |s ∅ ) )