Theorem max1ALT 13136

Description: A number is less than or equal to the maximum of it and another. This version of max1 13135 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13135 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 11240	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴)
2		iffalse 4470	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
3	2	breq2d 5091	. . 3 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ↔ 𝐴 ≤ 𝐴))
4	1, 3	syl5ibrcom 248	. 2 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)))
5		id 22	. . 3 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
6		iftrue 4467	. . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
7	5, 6	breqtrrd 5107	. 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 2119  ifcif 4461   class class class wbr 5079  ℝcr 11035   ≤ cle 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-pre-lttri 11110

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183

This theorem is referenced by: (None)