Theorem max1ALT 13186

Description: A number is less than or equal to the maximum of it and another. This version of max1 13185 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13185 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)

Assertion

Ref	Expression
max1ALT	⊢ (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

Proof of Theorem max1ALT

Step	Hyp	Ref	Expression
1		leid 11276	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴)
2		iffalse 4488	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
3	2	breq2d 5111	. . 3 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ↔ 𝐴 ≤ 𝐴))
4	1, 3	syl5ibrcom 249	. 2 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)))
5		id 22	. . 3 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
6		iftrue 4485	. . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
7	5, 6	breqtrrd 5127	. 2 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
8	4, 7	pm2.61d2 182	1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141  ifcif 4479   class class class wbr 5099  ℝcr 11069   ≤ cle 11214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-resscn 11127  ax-pre-lttri 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219

This theorem is referenced by: (None)