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Mirrors > Home > MPE Home > Th. List > elicc2 | Structured version Visualization version GIF version |
Description: Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elicc2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10085 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10085 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | elicc1 12219 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
4 | 1, 2, 3 | syl2an 494 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
5 | mnfxr 10096 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ*) |
7 | 1 | ad2antrr 762 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ*) |
8 | simpr1 1067 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ*) | |
9 | mnflt 11957 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
10 | 9 | ad2antrr 762 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐴) |
11 | simpr2 1068 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ≤ 𝐶) | |
12 | 6, 7, 8, 10, 11 | xrltletrd 11992 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶) |
13 | 2 | ad2antlr 763 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ*) |
14 | pnfxr 10092 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ*) |
16 | simpr3 1069 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵) | |
17 | ltpnf 11954 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
18 | 17 | ad2antlr 763 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞) |
19 | 8, 13, 15, 16, 18 | xrlelttrd 11991 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞) |
20 | xrrebnd 11999 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞))) | |
21 | 8, 20 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞))) |
22 | 12, 19, 21 | mpbir2and 957 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ) |
23 | 22, 11, 16 | 3jca 1242 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
24 | 23 | ex 450 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
25 | rexr 10085 | . . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
26 | 25 | 3anim1i 1248 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
27 | 24, 26 | impbid1 215 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
28 | 4, 27 | bitrd 268 | 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 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182 |
This theorem is referenced by: elicc2i 12239 iccssre 12255 iccsupr 12266 iccneg 12293 iccsplit 12305 iccshftr 12306 iccshftl 12308 iccdil 12310 icccntr 12312 iccf1o 12316 supicc 12320 icco1 14271 iccntr 22624 icccmplem1 22625 icccmplem2 22626 icccmplem3 22627 reconnlem1 22629 reconnlem2 22630 cnmpt2pc 22727 icoopnst 22738 iocopnst 22739 cnheiborlem 22753 ivthlem2 23221 ivthlem3 23222 ivthicc 23227 evthicc2 23229 ovolficc 23237 ovolicc1 23284 ovolicc2lem2 23286 ovolicc2lem5 23289 ovolicopnf 23292 dyadmaxlem 23365 opnmbllem 23369 volsup2 23373 volcn 23374 mbfi1fseqlem6 23487 itgspliticc 23603 itgsplitioo 23604 ditgcl 23622 ditgswap 23623 ditgsplitlem 23624 ditgsplit 23625 dvlip 23756 dvlip2 23758 dveq0 23763 dvgt0lem1 23765 dvivthlem1 23771 dvne0 23774 dvcnvrelem1 23780 dvcnvrelem2 23781 dvcnvre 23782 dvfsumlem2 23790 ftc1lem1 23798 ftc1lem2 23799 ftc1a 23800 ftc1lem4 23802 ftc2 23807 ftc2ditglem 23808 itgsubstlem 23811 pserulm 24176 loglesqrt 24499 log2tlbnd 24672 ppisval 24830 chtleppi 24935 fsumvma2 24939 chpchtsum 24944 chpub 24945 rplogsumlem2 25174 chpdifbndlem1 25242 pntibndlem2a 25279 pntibndlem2 25280 pntlemj 25292 pntlem3 25298 pntleml 25300 resconn 31228 cvmliftlem10 31276 opnmbllem0 33445 ftc2nc 33494 areacirclem2 33501 areacirclem4 33503 areacirc 33505 isbnd3 33583 isbnd3b 33584 prdsbnd 33592 iccbnd 33639 eliccd 39726 eliccre 39728 iccshift 39744 iccsuble 39745 limcicciooub 39869 icccncfext 40100 itgsubsticc 40192 iblcncfioo 40194 itgiccshift 40196 itgperiod 40197 itgsbtaddcnst 40198 fourierdlem42 40366 fourierdlem54 40377 fourierdlem63 40386 fourierdlem65 40388 fourierdlem74 40397 fourierdlem75 40398 fourierdlem82 40405 fourierdlem93 40416 fourierdlem101 40424 fourierdlem104 40427 fourierdlem111 40434 |
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