Theorem lesnltd 28047

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 28043	. 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 2145   class class class wbr 5102   No csur 27931   <s clts 27932   ≤s cles 28035

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-les 28036

This theorem is used by:  bdayfinbndlem1  28787