Theorem max1ALT 13230

Description: A number is less than or equal to the maximum of it and another. This version of max1 13229 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13229 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 11324	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴)
2		iffalse 4501	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
3	2	breq2d 5126	. . 3 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ↔ 𝐴 ≤ 𝐴))
4	1, 3	syl5ibrcom 250	. 2 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)))
5		id 23	. . 3 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
6		iftrue 4498	. . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
7	5, 6	breqtrrd 5144	. 2 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
8	4, 7	pm2.61d2 183	1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146  ifcif 4492   class class class wbr 5114  ℝcr 11117   ≤ cle 11262

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-resscn 11175  ax-pre-lttri 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267

This theorem is used by: (None)