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Theorem slenlt 32207
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 32200 . . . 4 ≤s = (( No × No ) ∖ <s )
21breqi 4861 . . 3 (𝐴 ≤s 𝐵𝐴(( No × No ) ∖ <s )𝐵)
3 brdif 4908 . . 3 (𝐴(( No × No ) ∖ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵))
4 brxp 5358 . . . 4 (𝐴( No × No )𝐵 ↔ (𝐴 No 𝐵 No ))
54anbi1i 612 . . 3 ((𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵) ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
62, 3, 53bitri 288 . 2 (𝐴 ≤s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
7 ibar 520 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵)))
8 brcnvg 5515 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐵 <s 𝐴))
98notbid 309 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
107, 9bitr3d 272 . 2 ((𝐴 No 𝐵 No ) → (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
116, 10syl5bb 274 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wcel 2157  cdif 3777   class class class wbr 4855   × cxp 5320  ccnv 5321   No csur 32123   <s cslt 32124   ≤s csle 32199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pr 5107
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856  df-opab 4918  df-xp 5328  df-cnv 5330  df-sle 32200
This theorem is referenced by:  sltnle  32208  sleloe  32209  sletri3  32210  sltletr  32211  slelttr  32212  sletr  32213  sltrec  32254
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