Theorem lesnltd 27896

Description: Surreal less-than or equal in terms of less-than. 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
lesnltd	⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Proof of Theorem lesnltd

Step	Hyp	Ref	Expression
1		lesd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		lesd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		lenlts 27892	. 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 2141   class class class wbr 5108   No csur 27780   <s clts 27781   ≤s cles 27884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-cnv 5669  df-les 27885

This theorem is referenced by:  bdayfinbndlem1  28636