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Theorem sleloe 32004
Description: Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
sleloe ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))

Proof of Theorem sleloe
StepHypRef Expression
1 slenlt 32002 . 2 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
2 orcom 401 . . . 4 ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 <s 𝐵))
3 eqcom 2658 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
43orbi1i 541 . . . 4 ((𝐴 = 𝐵𝐴 <s 𝐵) ↔ (𝐵 = 𝐴𝐴 <s 𝐵))
52, 4bitri 264 . . 3 ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ (𝐵 = 𝐴𝐴 <s 𝐵))
6 sltso 31952 . . . . . 6 <s Or No
7 sotric 5090 . . . . . 6 (( <s Or No ∧ (𝐵 No 𝐴 No )) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
86, 7mpan 706 . . . . 5 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
98ancoms 468 . . . 4 ((𝐴 No 𝐵 No ) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
109con2bid 343 . . 3 ((𝐴 No 𝐵 No ) → ((𝐵 = 𝐴𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
115, 10syl5bb 272 . 2 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ ¬ 𝐵 <s 𝐴))
121, 11bitr4d 271 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030   class class class wbr 4685   Or wor 5063   No csur 31918   <s cslt 31919   ≤s csle 31994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-sle 31995
This theorem is referenced by: (None)
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