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Theorem slenlt 32208
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 32201 . . . 4 ≤s = (( No × No ) ∖ <s )
21breqi 4790 . . 3 (𝐴 ≤s 𝐵𝐴(( No × No ) ∖ <s )𝐵)
3 brdif 4837 . . 3 (𝐴(( No × No ) ∖ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵))
4 brxp 5287 . . . 4 (𝐴( No × No )𝐵 ↔ (𝐴 No 𝐵 No ))
54anbi1i 602 . . 3 ((𝐴( No × No )𝐵 ∧ ¬ 𝐴 <s 𝐵) ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
62, 3, 53bitri 286 . 2 (𝐴 ≤s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵))
7 ibar 512 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵)))
8 brcnvg 5441 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐵 <s 𝐴))
98notbid 307 . . 3 ((𝐴 No 𝐵 No ) → (¬ 𝐴 <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
107, 9bitr3d 270 . 2 ((𝐴 No 𝐵 No ) → (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
116, 10syl5bb 272 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wcel 2144  cdif 3718   class class class wbr 4784   × cxp 5247  ccnv 5248   No csur 32124   <s cslt 32125   ≤s csle 32200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-sle 32201
This theorem is referenced by:  sltnle  32209  sleloe  32210  sletri3  32211  sltletr  32212  slelttr  32213  sletr  32214  sltrec  32255
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