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Theorem slenlt 27782
Description: Surreal less-than or equal in terms of less-than. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
slenlt ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Proof of Theorem slenlt
StepHypRef Expression
1 df-sle 27775 . . . 4 ≤s = (( No × No ) ∖ <s )
21breqi 5159 . . 3 (𝐴 ≤s 𝐵𝐴(( No × No ) ∖ <s )𝐵)
3 brdif 5206 . . 3 (𝐴(( No × No ) ∖ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵))
4 brxp 5731 . . . 4 (𝐴( No × No )𝐵 ↔ (𝐴 No 𝐵 No ))
54anbi1i 622 . . 3 ((𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵) ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
62, 3, 53bitri 296 . 2 (𝐴 ≤s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
7 ibar 527 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵)))
8 brcnvg 5886 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐵 <s 𝐴))
98notbid 317 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
107, 9bitr3d 280 . 2 ((𝐴 No 𝐵 No ) → (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
116, 10bitrid 282 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2099  cdif 3944   class class class wbr 5153   × cxp 5680  ccnv 5681   No csur 27669   <s cslt 27670   ≤s csle 27774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-sle 27775
This theorem is referenced by:  sltnle  27783  sleloe  27784  sletri3  27785  sltletr  27786  slelttr  27787  sletr  27788  slerflex  27793  sletric  27794  sltled  27799  sltrec  27850  sltlpss  27930  cofcutr  27941  sleneg  28055  slesubsubbd  28093  slesubsub2bd  28094  slesubsub3bd  28095  slesubaddd  28100  slemul2d  28175  slemul1d  28176  sltonold  28254  om2noseqlt2  28274
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