Theorem ltnegsim

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

	Ref	Expression
Assertion	ltnegsim	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 → ( -us ‘ 𝐵 ) <s ( -us ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negsprop	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( 𝐴 <s 𝐵 → ( -us ‘ 𝐵 ) <s ( -us ‘ 𝐴 ) ) ) )
2	1	simprd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 → ( -us ‘ 𝐵 ) <s ( -us ‘ 𝐴 ) ) )