Theorem sltnegim

Description: The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	sltnegim	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\to {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		negsprop	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left({+}_s\left(A\right)\in \mathrm{No}\wedge \left(A{<}_{s}B\to {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)\right)$$
2	1	simprd	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\to {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)$$