Theorem negcut

Description: The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negcut	⊢ ( 𝐴 ∈ No → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } ∧ { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		negsprop	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( ( -us ‘ 𝑥 ) ∈ No ∧ ( 𝑥 <s 𝑦 → ( -us ‘ 𝑦 ) <s ( -us ‘ 𝑥 ) ) ) )
2	1	a1d	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( ( ( bday ‘ 𝑥 ) ∪ ( bday ‘ 𝑦 ) ) ∈ ( ( bday ‘ 𝐴 ) ∪ ( bday ‘ 0s ) ) → ( ( -us ‘ 𝑥 ) ∈ No ∧ ( 𝑥 <s 𝑦 → ( -us ‘ 𝑦 ) <s ( -us ‘ 𝑥 ) ) ) ) )
3	2	rgen2	⊢ ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ( ( ( bday ‘ 𝑥 ) ∪ ( bday ‘ 𝑦 ) ) ∈ ( ( bday ‘ 𝐴 ) ∪ ( bday ‘ 0s ) ) → ( ( -us ‘ 𝑥 ) ∈ No ∧ ( 𝑥 <s 𝑦 → ( -us ‘ 𝑦 ) <s ( -us ‘ 𝑥 ) ) ) )
4	3	a1i	⊢ ( 𝐴 ∈ No → ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ( ( ( bday ‘ 𝑥 ) ∪ ( bday ‘ 𝑦 ) ) ∈ ( ( bday ‘ 𝐴 ) ∪ ( bday ‘ 0s ) ) → ( ( -us ‘ 𝑥 ) ∈ No ∧ ( 𝑥 <s 𝑦 → ( -us ‘ 𝑦 ) <s ( -us ‘ 𝑥 ) ) ) ) )
5		id	⊢ ( 𝐴 ∈ No → 𝐴 ∈ No )
6	4 5	negsproplem3	⊢ ( 𝐴 ∈ No → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( -us “ ( R ‘ 𝐴 ) ) <<s { ( -us ‘ 𝐴 ) } ∧ { ( -us ‘ 𝐴 ) } <<s ( -us “ ( L ‘ 𝐴 ) ) ) )