Theorem iocval 13444

Description: Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
iocval	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iocval

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioc 13412	. 2 ⊢ (,] = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦 < 𝑥 ∧ 𝑥 ≤ 𝑧)})
2	1	ixxval 13415	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443   class class class wbr 5166  (class class class)co 7448  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  (,]cioc 13408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-xr 11328  df-ioc 13412

This theorem is referenced by:  ioc0  13454  orvclteel  34437