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Mirrors > Home > MPE Home > Th. List > elioc2 | Structured version Visualization version GIF version |
Description: Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elioc2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10675 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
2 | elioc1 12768 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
3 | 1, 2 | sylan2 592 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
4 | mnfxr 10686 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ*) |
6 | simpll 763 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ*) | |
7 | simpr1 1186 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ*) | |
8 | mnfle 12517 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
9 | 8 | ad2antrr 722 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ≤ 𝐴) |
10 | simpr2 1187 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 < 𝐶) | |
11 | 5, 6, 7, 9, 10 | xrlelttrd 12541 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶) |
12 | 1 | ad2antlr 723 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 10683 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ*) |
15 | simpr3 1188 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵) | |
16 | ltpnf 12503 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
17 | 16 | ad2antlr 723 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞) |
18 | 7, 12, 14, 15, 17 | xrlelttrd 12541 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞) |
19 | xrrebnd 12549 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞))) | |
20 | 7, 19 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞))) |
21 | 11, 18, 20 | mpbir2and 709 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ) |
22 | 21, 10, 15 | 3jca 1120 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
23 | 22 | ex 413 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
24 | rexr 10675 | . . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
25 | 24 | 3anim1i 1144 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
26 | 23, 25 | impbid1 226 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
27 | 3, 26 | bitrd 280 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 (,]cioc 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioc 12731 |
This theorem is referenced by: iocssre 12804 ef01bndlem 15525 sin01bnd 15526 cos01bnd 15527 cos1bnd 15528 sinltx 15530 sin01gt0 15531 cos01gt0 15532 sin02gt0 15533 sincos1sgn 15534 sincos2sgn 15535 icoopnst 23470 iocopnst 23471 ismbf3d 24182 aaliou3lem2 24859 aaliou3lem3 24860 pilem2 24967 sinhalfpilem 24976 sincosq1lem 25010 coseq0negpitopi 25016 tangtx 25018 sincos4thpi 25026 efif1olem1 25053 efif1olem2 25054 efif1o 25057 efifo 25058 ellogrn 25070 logimclad 25083 ellogdm 25149 logdmnrp 25151 dvloglem 25158 dvlog2lem 25162 asinneg 25391 atans2 25436 ressatans 25439 abvcxp 26118 ostth2 26140 xrge0iifcv 31076 xrge0iifiso 31077 xrge0iifhom 31079 sinccvglem 32812 bj-pinftyccb 34395 bj-pinftynminfty 34401 dvasin 34859 areacirclem4 34866 gtnelioc 41641 limcicciooub 41794 fourierdlem4 42273 fourierdlem26 42295 fourierdlem33 42302 fourierdlem37 42306 fourierdlem65 42333 fourierdlem79 42347 fouriersw 42393 eenglngeehlnmlem1 44652 eenglngeehlnmlem2 44653 |
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