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Theorem slenlt 27052
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 27045 . . . 4 ≤s = (( No × No ) ∖ <s )
21breqi 5109 . . 3 (𝐴 ≤s 𝐵𝐴(( No × No ) ∖ <s )𝐵)
3 brdif 5156 . . 3 (𝐴(( No × No ) ∖ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵))
4 brxp 5679 . . . 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 296 . 2 (𝐴 ≤s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
7 ibar 529 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵)))
8 brcnvg 5833 . . . 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 396  wcel 2106  cdif 3905   class class class wbr 5103   × cxp 5629  ccnv 5630   No csur 26940   <s cslt 26941   ≤s csle 27044
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-sle 27045
This theorem is referenced by:  sltnle  27053  sleloe  27054  sletri3  27055  sltletr  27056  slelttr  27057  sletr  27058  slerflex  27063  sltrec  27111  sltlpss  27185  cofcutr  27192  sleneg  34338
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