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Theorem max1ALT 13104
Description: A number is less than or equal to the maximum of it and another. This version of max1 13103 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13103 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 11250 . . 3 (𝐴 ∈ ℝ → 𝐴𝐴)
2 iffalse 4495 . . . 4 𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐴)
32breq2d 5117 . . 3 𝐴𝐵 → (𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴) ↔ 𝐴𝐴))
41, 3syl5ibrcom 246 . 2 (𝐴 ∈ ℝ → (¬ 𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴)))
5 id 22 . . 3 (𝐴𝐵𝐴𝐵)
6 iftrue 4492 . . 3 (𝐴𝐵 → if(𝐴𝐵, 𝐵, 𝐴) = 𝐵)
75, 6breqtrrd 5133 . 2 (𝐴𝐵𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
84, 7pm2.61d2 181 1 (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  ifcif 4486   class class class wbr 5105  cr 11049  cle 11189
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-resscn 11107  ax-pre-lttri 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8647  df-en 8883  df-dom 8884  df-sdom 8885  df-pnf 11190  df-mnf 11191  df-xr 11192  df-ltxr 11193  df-le 11194
This theorem is referenced by: (None)
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