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Theorem sltnle 33340
Description: Surreal less than in terms of less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
sltnle ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Proof of Theorem sltnle
StepHypRef Expression
1 slenlt 33339 . . 3 ((𝐵 No 𝐴 No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
21ancoms 462 . 2 ((𝐴 No 𝐵 No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
32con2bid 358 1 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112   class class class wbr 5033   No csur 33255   <s cslt 33256   ≤s csle 33331
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-sle 33332
This theorem is referenced by:  slerec  33385  sltrec  33386
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