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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 10695 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | elioo2 12780 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) |
4 | df-3an 1085 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) | |
5 | ltpnf 12516 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 < +∞) |
7 | 6 | pm4.71i 562 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) |
8 | 4, 7 | bitr4i 280 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
9 | 3, 8 | syl6bb 289 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 (,)cioo 12739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 |
This theorem is referenced by: mbfmulc2lem 24248 mbfposr 24253 ismbf3d 24255 mbfaddlem 24261 mbfsup 24265 itg2gt0 24361 itg2cnlem1 24362 itg2cnlem2 24363 lhop2 24612 dvfsumlem2 24624 dvfsumlem3 24625 dvfsumrlimge0 24627 dvfsumrlim 24628 dvfsumrlim2 24629 pntpbnd1a 26161 pntpbnd2 26163 pntibndlem2 26167 pntibndlem3 26168 pntlemi 26180 pntlemo 26183 relowlssretop 34647 itg2addnclem2 34959 iblabsnclem 34970 ftc1anclem1 34982 ftc1anclem6 34987 rfcnpre1 41325 regt1loggt0 44645 rege1logbrege0 44667 rege1logbzge0 44668 |
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