Theorem ltnles 28119

Description: Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
ltnles	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Proof of Theorem ltnles

Step	Hyp	Ref	Expression
1		lenlts 28118	. . 3 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
2	1	ancoms 464	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
3	2	con2bid 357	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144   class class class wbr 5102  Nocsur 28006   <s clts 28007   ≤s cles 28110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-cnv 5659  df-les 28111

This theorem is used by:  ltsnled  28123  lestric  28134  lesrec  28194  ltsrec  28196  eqcuts3  28199  cutlt  28327  leadds1im  28382  leadds1  28384  ltadds2  28386  abssnid  28638