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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 11312 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | elico2 13447 | . . 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 13159 | . . . . 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 1088 | . . 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 1086 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 ℝcr 11151 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 [,)cico 13385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-ico 13389 |
This theorem is referenced by: elrege0 13490 rexico 15388 limsupgle 15509 limsupgre 15513 rlim3 15530 ello12 15548 lo1bdd2 15556 elo12 15559 lo1resb 15596 rlimresb 15597 o1resb 15598 lo1eq 15600 rlimeq 15601 rlimsqzlem 15681 o1fsum 15845 ovolicopnf 25572 dvfsumrlimge0 26085 dvfsumrlim 26086 dvfsumrlim2 26087 cxp2lim 27034 chebbnd1 27530 chtppilimlem1 27531 chtppilimlem2 27532 chtppilim 27533 chebbnd2 27535 chto1lb 27536 chpchtlim 27537 chpo1ub 27538 vmadivsumb 27541 dchrisumlema 27546 dchrisumlem2 27548 dchrisumlem3 27549 dchrmusumlema 27551 dchrmusum2 27552 dchrvmasumlem2 27556 dchrvmasumiflem1 27559 dchrisum0lema 27572 dchrisum0lem1b 27573 dchrisum0lem2a 27575 dchrisum0lem2 27576 2vmadivsumlem 27598 selbergb 27607 selberg2b 27610 chpdifbndlem1 27611 selberg3lem1 27615 selberg3lem2 27616 selberg4lem1 27618 pntrsumo1 27623 selbergsb 27633 pntrlog2bndlem3 27637 pntpbnd1 27644 pntpbnd2 27645 pntibndlem3 27650 pntlemn 27658 pntlem3 27667 pntleml 27669 pnt2 27671 uzssico 32792 itg2addnclem2 37658 2xp3dxp2ge1d 42222 elbigo2 48401 rege1logbrege0 48407 blennnelnn 48425 dignnld 48452 |
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