Theorem ltsnled 27902

Description: Surreal less-than in terms of less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)

Hypotheses

Ref	Expression
lesd.1	⊢ (𝜑 → 𝐴 ∈ No )
lesd.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
ltsnled	⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Proof of Theorem ltsnled

Step	Hyp	Ref	Expression
1		lesd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		lesd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		ltnles 27898	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2143   class class class wbr 5110   No csur 27785   <s clts 27786   ≤s cles 27889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-les 27890

This theorem is referenced by:  bdayfinbndlem1  28641