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Theorem maxs2 27603
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 27600 . . . 4 (𝐵 No 𝐵 ≤s 𝐵)
21ad2antlr 724 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → 𝐵 ≤s 𝐵)
3 iftrue 4526 . . . 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 5166 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
6 sletric 27601 . . . 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 4529 . . . 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 5166 . 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 4520   class class class wbr 5138   No csur 27477   ≤s csle 27581
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 5289  ax-nul 5296  ax-pr 5417
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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-1o 8461  df-2o 8462  df-no 27480  df-slt 27481  df-sle 27582
This theorem is referenced by: (None)
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