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Theorem maxs2 27639
Description: A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
Assertion
Ref Expression
maxs2 ((𝐴 No 𝐵 No ) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))

Proof of Theorem maxs2
StepHypRef Expression
1 slerflex 27636 . . . 4 (𝐵 No 𝐵 ≤s 𝐵)
21ad2antlr 724 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → 𝐵 ≤s 𝐵)
3 iftrue 4527 . . . 4 (𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐵, 𝐴) = 𝐵)
43adantl 481 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐵, 𝐴) = 𝐵)
52, 4breqtrrd 5167 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
6 sletric 27637 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵𝐵 ≤s 𝐴))
76orcanai 999 . . 3 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → 𝐵 ≤s 𝐴)
8 iffalse 4530 . . . 4 𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐵, 𝐴) = 𝐴)
98adantl 481 . . 3 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐵, 𝐴) = 𝐴)
107, 9breqtrrd 5167 . 2 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
115, 10pm2.61dan 810 1 ((𝐴 No 𝐵 No ) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  ifcif 4521   class class class wbr 5139   No csur 27513   ≤s csle 27617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-1o 8462  df-2o 8463  df-no 27516  df-slt 27517  df-sle 27618
This theorem is referenced by: (None)
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