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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 11210 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | elioo2 13306 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞))) |
4 | df-3an 1090 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) | |
5 | ltpnf 13042 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
6 | 5 | adantr 482 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 < +∞) |
7 | 6 | pm4.71i 561 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞)) |
8 | 4, 7 | bitr4i 278 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
9 | 3, 8 | bitrdi 287 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11051 +∞cpnf 11187 ℝ*cxr 11189 < clt 11190 (,)cioo 13265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ioo 13269 |
This theorem is referenced by: mbfmulc2lem 25014 mbfposr 25019 ismbf3d 25021 mbfaddlem 25027 mbfsup 25031 itg2gt0 25128 itg2cnlem1 25129 itg2cnlem2 25130 lhop2 25382 dvfsumlem2 25394 dvfsumlem3 25395 dvfsumrlimge0 25397 dvfsumrlim 25398 dvfsumrlim2 25399 pntpbnd1a 26936 pntpbnd2 26938 pntibndlem2 26942 pntibndlem3 26943 pntlemi 26955 pntlemo 26958 relowlssretop 35837 itg2addnclem2 36133 iblabsnclem 36144 ftc1anclem1 36154 ftc1anclem6 36159 rfcnpre1 43231 regt1loggt0 46629 rege1logbrege0 46651 rege1logbzge0 46652 io1ii 46960 |
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