Theorem sltne

Description: Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Assertion	sltne	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 <s 𝐵 ) → 𝐵 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sltirr	⊢ ( 𝐴 ∈ No → ¬ 𝐴 <s 𝐴 )
2		breq2	⊢ ( 𝐵 = 𝐴 → ( 𝐴 <s 𝐵 ↔ 𝐴 <s 𝐴 ) )
3	2	notbid	⊢ ( 𝐵 = 𝐴 → ( ¬ 𝐴 <s 𝐵 ↔ ¬ 𝐴 <s 𝐴 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐴 ∈ No → ( 𝐵 = 𝐴 → ¬ 𝐴 <s 𝐵 ) )
5	4	necon2ad	⊢ ( 𝐴 ∈ No → ( 𝐴 <s 𝐵 → 𝐵 ≠ 𝐴 ) )
6	5	imp	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 <s 𝐵 ) → 𝐵 ≠ 𝐴 )