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Theorem max1ALT 12572
Description: A number is less than or equal to the maximum of it and another. This version of max1 12571 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 12571 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)
Assertion
Ref Expression
max1ALT (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Proof of Theorem max1ALT
StepHypRef Expression
1 leid 10728 . . 3 (𝐴 ∈ ℝ → 𝐴𝐴)
2 iffalse 4458 . . . 4 𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐴)
32breq2d 5064 . . 3 𝐴𝐵 → (𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴) ↔ 𝐴𝐴))
41, 3syl5ibrcom 250 . 2 (𝐴 ∈ ℝ → (¬ 𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴)))
5 id 22 . . 3 (𝐴𝐵𝐴𝐵)
6 iftrue 4455 . . 3 (𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐵)
75, 6breqtrrd 5080 . 2 (𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
84, 7pm2.61d2 184 1 (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2115  ifcif 4449   class class class wbr 5052  cr 10528  cle 10668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-resscn 10586  ax-pre-lttri 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673
This theorem is referenced by: (None)
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