Theorem sltnled 27727

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

Hypotheses

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

Assertion

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

Proof of Theorem sltnled

Step	Hyp	Ref	Expression
1		sled.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		sled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		sltnle 27723	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2114   class class class wbr 5097   No csur 27609   <s cslt 27610   ≤s csle 27714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-cnv 5631  df-sle 27715

This theorem is referenced by:  bdayfinbndlem1  28444