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Theorem slenlt 27797
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 27790 . . . 4 ≤s = (( No × No ) ∖ <s )
21breqi 5149 . . 3 (𝐴 ≤s 𝐵𝐴(( No × No ) ∖ <s )𝐵)
3 brdif 5196 . . 3 (𝐴(( No × No ) ∖ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵))
4 brxp 5734 . . . 4 (𝐴( No × No )𝐵 ↔ (𝐴 No 𝐵 No ))
54anbi1i 624 . . 3 ((𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵) ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
62, 3, 53bitri 297 . 2 (𝐴 ≤s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
7 ibar 528 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵)))
8 brcnvg 5890 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐵 <s 𝐴))
98notbid 318 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
107, 9bitr3d 281 . 2 ((𝐴 No 𝐵 No ) → (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
116, 10bitrid 283 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2108  cdif 3948   class class class wbr 5143   × cxp 5683  ccnv 5684   No csur 27684   <s cslt 27685   ≤s csle 27789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-sle 27790
This theorem is referenced by:  sltnle  27798  sleloe  27799  sletri3  27800  sltletr  27801  slelttr  27802  sletr  27803  slerflex  27808  sletric  27809  sltled  27814  sltrec  27865  sltlpss  27945  cofcutr  27958  sleneg  28078  slesubsubbd  28116  slesubsub2bd  28117  slesubsub3bd  28118  slesubaddd  28123  slemul2d  28200  slemul1d  28201  sltonold  28283  om2noseqlt2  28306
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