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Theorem mins1 27699
Description: The minimum of two surreals is less than or equal to the first. (Contributed by Scott Fenton, 14-Feb-2025.)
Assertion
Ref Expression
mins1 ((𝐴 No 𝐵 No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)

Proof of Theorem mins1
StepHypRef Expression
1 iftrue 4535 . . . 4 (𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐴)
21adantl 481 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐴)
3 slerflex 27695 . . . 4 (𝐴 No 𝐴 ≤s 𝐴)
43ad2antrr 725 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → 𝐴 ≤s 𝐴)
52, 4eqbrtrd 5170 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)
6 iffalse 4538 . . . 4 𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐵)
76adantl 481 . . 3 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐵)
8 sletric 27696 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵𝐵 ≤s 𝐴))
98orcanai 1001 . . 3 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → 𝐵 ≤s 𝐴)
107, 9eqbrtrd 5170 . 2 (((𝐴 No 𝐵 No ) ∧ ¬ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)
115, 10pm2.61dan 812 1 ((𝐴 No 𝐵 No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  ifcif 4529   class class class wbr 5148   No csur 27572   ≤s csle 27676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-1o 8486  df-2o 8487  df-no 27575  df-slt 27576  df-sle 27677
This theorem is referenced by: (None)
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