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Theorem max1ALT 11891
Description: A number is less than or equal to the maximum of it and another. This version of max1 11890 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 11890 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 10012 . . 3 (𝐴 ∈ ℝ → 𝐴𝐴)
2 iffalse 4045 . . . 4 𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐴)
32breq2d 4595 . . 3 𝐴𝐵 → (𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴) ↔ 𝐴𝐴))
41, 3syl5ibrcom 236 . 2 (𝐴 ∈ ℝ → (¬ 𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴)))
5 id 22 . . 3 (𝐴𝐵𝐴𝐵)
6 iftrue 4042 . . 3 (𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐵)
75, 6breqtrrd 4611 . 2 (𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
84, 7pm2.61d2 171 1 (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1977  ifcif 4036   class class class wbr 4583  cr 9814  cle 9954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-pre-lttri 9889
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959
This theorem is referenced by: (None)
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