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Mirrors > Home > MPE Home > Th. List > elioopnf | Structured version Visualization version GIF version |
Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
elioopnf | ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 10292 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | elioo2 12414 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) | |
3 | 1, 2 | mpan2 671 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) |
4 | df-3an 1073 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) | |
5 | ltpnf 12152 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
6 | 5 | adantr 466 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 < +∞) |
7 | 6 | pm4.71i 549 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) |
8 | 4, 7 | bitr4i 267 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
9 | 3, 8 | syl6bb 276 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6791 ℝcr 10135 +∞cpnf 10271 ℝ*cxr 10273 < clt 10274 (,)cioo 12373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-pre-lttri 10210 ax-pre-lttrn 10211 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-1st 7313 df-2nd 7314 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-ioo 12377 |
This theorem is referenced by: mbfmulc2lem 23627 mbfposr 23632 ismbf3d 23634 mbfaddlem 23640 mbfsup 23644 itg2gt0 23740 itg2cnlem1 23741 itg2cnlem2 23742 lhop2 23991 dvfsumlem2 24003 dvfsumlem3 24004 dvfsumrlimge0 24006 dvfsumrlim 24007 dvfsumrlim2 24008 pntpbnd1a 25488 pntpbnd2 25490 pntibndlem2 25494 pntibndlem3 25495 pntlemi 25507 pntlemo 25510 relowlssretop 33541 itg2addnclem2 33787 iblabsnclem 33798 ftc1anclem1 33810 ftc1anclem6 33815 rfcnpre1 39693 regt1loggt0 42851 rege1logbrege0 42873 rege1logbzge0 42874 |
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