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| 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 11315 | . . 3 ⊢ +∞ ∈ ℝ* | |
| 2 | elico2 13451 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞))) | |
| 3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞))) |
| 4 | ltpnf 13162 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
| 5 | 4 | adantr 480 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 < +∞) |
| 6 | 5 | pm4.71i 559 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ∧ 𝐵 < +∞)) |
| 7 | df-3an 1089 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ∧ 𝐵 < +∞)) | |
| 8 | 6, 7 | bitr4i 278 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞)) |
| 9 | 3, 8 | bitr4di 289 | 1 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 |
| This theorem is referenced by: elrege0 13494 rexico 15392 limsupgle 15513 limsupgre 15517 rlim3 15534 ello12 15552 lo1bdd2 15560 elo12 15563 lo1resb 15600 rlimresb 15601 o1resb 15602 lo1eq 15604 rlimeq 15605 rlimsqzlem 15685 o1fsum 15849 ovolicopnf 25559 dvfsumrlimge0 26071 dvfsumrlim 26072 dvfsumrlim2 26073 cxp2lim 27020 chebbnd1 27516 chtppilimlem1 27517 chtppilimlem2 27518 chtppilim 27519 chebbnd2 27521 chto1lb 27522 chpchtlim 27523 chpo1ub 27524 vmadivsumb 27527 dchrisumlema 27532 dchrisumlem2 27534 dchrisumlem3 27535 dchrmusumlema 27537 dchrmusum2 27538 dchrvmasumlem2 27542 dchrvmasumiflem1 27545 dchrisum0lema 27558 dchrisum0lem1b 27559 dchrisum0lem2a 27561 dchrisum0lem2 27562 2vmadivsumlem 27584 selbergb 27593 selberg2b 27596 chpdifbndlem1 27597 selberg3lem1 27601 selberg3lem2 27602 selberg4lem1 27604 pntrsumo1 27609 selbergsb 27619 pntrlog2bndlem3 27623 pntpbnd1 27630 pntpbnd2 27631 pntibndlem3 27636 pntlemn 27644 pntlem3 27653 pntleml 27655 pnt2 27657 uzssico 32786 itg2addnclem2 37679 2xp3dxp2ge1d 42242 elbigo2 48473 rege1logbrege0 48479 blennnelnn 48497 dignnld 48524 |
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