Theorem ltsnled 28147

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 28143	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2147   class class class wbr 5106  Nocsur 28030   <s clts 28031   ≤s cles 28134

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-cnv 5664  df-les 28135

This theorem is used by:  bdayfinbndlem1  28886