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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoreelrn | Structured version Visualization version GIF version |
Description: A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.) |
Ref | Expression |
---|---|
icoreelrn.1 | ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) |
Ref | Expression |
---|---|
icoreelrn | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoreval 35524 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | |
2 | simpl 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
3 | simpr 485 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
4 | df-ico 13085 | . . . . . 6 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
5 | 4 | ixxf 13089 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
6 | ffun 6603 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
7 | 5, 6 | mp1i 13 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → Fun [,)) |
8 | rexpssxrxp 11020 | . . . . . 6 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
9 | 5 | fdmi 6612 | . . . . . 6 ⊢ dom [,) = (ℝ* × ℝ*) |
10 | 8, 9 | sseqtrri 3958 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ dom [,) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ × ℝ) ⊆ dom [,)) |
12 | 2, 3, 7, 11 | elovimad 7323 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ ([,) “ (ℝ × ℝ))) |
13 | icoreelrn.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
14 | 12, 13 | eleqtrrdi 2850 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ 𝐼) |
15 | 1, 14 | eqeltrrd 2840 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 𝒫 cpw 4533 class class class wbr 5074 × cxp 5587 dom cdm 5589 “ cima 5592 Fun wfun 6427 ⟶wf 6429 (class class class)co 7275 ℝcr 10870 ℝ*cxr 11008 < clt 11009 ≤ cle 11010 [,)cico 13081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ico 13085 |
This theorem is referenced by: relowlssretop 35534 |
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