Theorem lesnltd 27720

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 27716	. 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 5085   No csur 27603   <s clts 27604   ≤s cles 27708

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-les 27709

This theorem is referenced by:  bdayfinbndlem1  28459