Theorem lt0negs2d

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

	Ref	Expression
Hypothesis	lt0negs2d.1	$$⊢\phi \to A\in \mathrm{No}$$
Assertion	lt0negs2d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow {+}_s\left(A\right){<}_s{0}_s\right)$$

Proof

Step	Hyp	Ref	Expression
1		lt0negs2d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		0no	$$⊢0_s\in \mathrm{No}$$
3		ltnegs	$$⊢\left(0_s\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(0_s{<}_{s}A\leftrightarrow {+}_s\left(A\right){<}_s{+}_s\left(0_s\right)\right)$$
4	2 1 3	sylancr	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow {+}_s\left(A\right){<}_s{+}_s\left(0_s\right)\right)$$
5		neg0s	$$⊢{+}_s\left(0_s\right)=0_s$$
6	5	breq2i	$$⊢{+}_s\left(A\right){<}_s{+}_s\left(0_s\right)\leftrightarrow {+}_s\left(A\right){<}_s{0}_s$$
7	4 6	bitrdi	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow {+}_s\left(A\right){<}_s{0}_s\right)$$