Theorem lt0negs2d

Description: Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypothesis	lt0negs2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
Assertion	lt0negs2d	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s 0s ) )

Proof

Step	Hyp	Ref	Expression
1		lt0negs2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		0no	⊢ 0s ∈ No
3		ltnegs	⊢ ( ( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s ( -us ‘ 0s ) ) )
4	2 1 3	sylancr	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s ( -us ‘ 0s ) ) )
5		neg0s	⊢ ( -us ‘ 0s ) = 0s
6	5	breq2i	⊢ ( ( -us ‘ 𝐴 ) <s ( -us ‘ 0s ) ↔ ( -us ‘ 𝐴 ) <s 0s )
7	4 6	bitrdi	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s 0s ) )