![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elicopnf | Structured version Visualization version GIF version |
Description: Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.) |
Ref | Expression |
---|---|
elicopnf | ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 10417 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | elico2 12532 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞))) | |
3 | 1, 2 | mpan2 682 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞))) |
4 | ltpnf 12247 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 < +∞) |
6 | 5 | pm4.71i 555 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ∧ 𝐵 < +∞)) |
7 | df-3an 1113 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ∧ 𝐵 < +∞)) | |
8 | 6, 7 | bitr4i 270 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞)) |
9 | 3, 8 | syl6bbr 281 | 1 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 ∈ wcel 2164 class class class wbr 4875 (class class class)co 6910 ℝcr 10258 +∞cpnf 10395 ℝ*cxr 10397 < clt 10398 ≤ cle 10399 [,)cico 12472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-ico 12476 |
This theorem is referenced by: elrege0 12575 rexico 14477 limsupgle 14592 limsupgre 14596 rlim3 14613 ello12 14631 lo1bdd2 14639 elo12 14642 lo1resb 14679 rlimresb 14680 o1resb 14681 lo1eq 14683 rlimeq 14684 rlimsqzlem 14763 o1fsum 14926 ovolicopnf 23697 dvfsumrlimge0 24199 dvfsumrlim 24200 dvfsumrlim2 24201 cxp2lim 25123 chebbnd1 25581 chtppilimlem1 25582 chtppilimlem2 25583 chtppilim 25584 chebbnd2 25586 chto1lb 25587 chpchtlim 25588 chpo1ub 25589 vmadivsumb 25592 dchrisumlema 25597 dchrisumlem2 25599 dchrisumlem3 25600 dchrmusumlema 25602 dchrmusum2 25603 dchrvmasumlem2 25607 dchrvmasumiflem1 25610 dchrisum0lema 25623 dchrisum0lem1b 25624 dchrisum0lem2a 25626 dchrisum0lem2 25627 2vmadivsumlem 25649 selbergb 25658 selberg2b 25661 chpdifbndlem1 25662 selberg3lem1 25666 selberg3lem2 25667 selberg4lem1 25669 pntrsumo1 25674 selbergsb 25684 pntrlog2bndlem3 25688 pntpbnd1 25695 pntpbnd2 25696 pntibndlem3 25701 pntlemn 25709 pntlem3 25718 pntleml 25720 pnt2 25722 uzssico 30089 itg2addnclem2 34004 elbigo2 43211 rege1logbrege0 43217 blennnelnn 43235 dignnld 43262 |
Copyright terms: Public domain | W3C validator |