Theorem negscut2

Description: The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negscut2	⊢ ( 𝐴 ∈ No → ( -us “ ( R ‘ 𝐴 ) ) <<s ( -us “ ( L ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negscut	⊢ ( 𝐴 ∈ No → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } ∧ { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) ) )
2	1	simp2d	⊢ ( 𝐴 ∈ No → ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } )
3	1	simp3d	⊢ ( 𝐴 ∈ No → { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) )
4		fvex	⊢ ( -us ‘ 𝐴 ) ∈ V
5	4	snnz	⊢ { ( -us ‘ 𝐴 ) } ≠ ∅
6		sslttr	⊢ ( ( ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } ∧ { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) ∧ { ( -us ‘ 𝐴 ) } ≠ ∅ ) → ( -us “ ( R ‘ 𝐴 ) ) <<s ( -us “ ( L ‘ 𝐴 ) ) )
7	5 6	mp3an3	⊢ ( ( ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } ∧ { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) ) → ( -us “ ( R ‘ 𝐴 ) ) <<s ( -us “ ( L ‘ 𝐴 ) ) )
8	2 3 7	syl2anc	⊢ ( 𝐴 ∈ No → ( -us “ ( R ‘ 𝐴 ) ) <<s ( -us “ ( L ‘ 𝐴 ) ) )