Theorem ltsm1d

Description: A surreal is greater than itself minus one. (Contributed by Scott Fenton, 20-Aug-2025)

	Ref	Expression
Hypothesis	ltsm1d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
Assertion	ltsm1d	⊢ ( 𝜑 → ( 𝐴 -s 1s ) <s 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltsm1d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2	1	ltsp1d	⊢ ( 𝜑 → 𝐴 <s ( 𝐴 +s 1s ) )
3		1no	⊢ 1s ∈ No
4	3	a1i	⊢ ( 𝜑 → 1s ∈ No )
5	1 4 1	ltsubaddsd	⊢ ( 𝜑 → ( ( 𝐴 -s 1s ) <s 𝐴 ↔ 𝐴 <s ( 𝐴 +s 1s ) ) )
6	2 5	mpbird	⊢ ( 𝜑 → ( 𝐴 -s 1s ) <s 𝐴 )