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Mirrors > Home > MPE Home > Th. List > ltpnf | Structured version Visualization version GIF version |
Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ltpnf | ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ +∞ = +∞ | |
2 | orc 866 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) | |
3 | 1, 2 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) |
4 | 3 | olcd 873 | . 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))) |
5 | rexr 11202 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
6 | pnfxr 11210 | . . 3 ⊢ +∞ ∈ ℝ* | |
7 | ltxr 13037 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancl 587 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) |
9 | 4, 8 | mpbird 257 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ℝcr 11051 <ℝ cltrr 11056 +∞cpnf 11187 -∞cmnf 11188 ℝ*cxr 11189 < clt 11190 |
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-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 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-br 5107 df-opab 5169 df-xp 5640 df-pnf 11192 df-xr 11194 df-ltxr 11195 |
This theorem is referenced by: ltpnfd 13043 0ltpnf 13044 xrlttri 13059 xrlttr 13060 xrrebnd 13088 xrre 13089 qbtwnxr 13120 xnn0lem1lt 13164 xrinfmsslem 13228 xrub 13232 supxrunb1 13239 supxrunb2 13240 dfrp2 13314 elioc2 13328 elicc2 13330 ioomax 13340 ioopos 13342 elioopnf 13361 elicopnf 13363 difreicc 13402 hashbnd 14237 hashv01gt1 14246 fprodge0 15877 fprodge1 15879 pcadd 16762 ramubcl 16891 rge0srg 20871 mnfnei 22575 icopnfcld 24134 iocmnfcld 24135 xrtgioo 24172 xrge0tsms 24200 ioombl1lem4 24928 icombl1 24930 mbfmax 25016 upgrfi 28045 topnfbey 29416 isblo3i 29746 htthlem 29862 xlt2addrd 31666 fsumrp0cl 31889 xrge0tsmsd 31902 pnfinf 32022 xrge0slmod 32143 xrge0iifcnv 32517 xrge0iifiso 32519 xrge0iifhom 32521 lmxrge0 32536 esumcst 32665 esumcvgre 32693 voliune 32831 volfiniune 32832 sxbrsigalem0 32874 orvcgteel 33070 dstfrvclim1 33080 itg2addnclem2 36133 asindmre 36164 dvasin 36165 dvacos 36166 rfcnpre3 43245 supxrgere 43574 supxrgelem 43578 xrlexaddrp 43593 infxr 43608 xrpnf 43728 limsupre 43889 limsuppnflem 43958 liminflelimsupuz 44033 limsupub2 44060 icccncfext 44135 fourierdlem111 44465 fourierdlem113 44467 fouriersw 44479 sge0iunmptlemre 44663 sge0rpcpnf 44669 sge0xaddlem1 44681 meaiuninclem 44728 hoidmvlelem5 44847 ovolval5lem1 44900 pimltpnff 44951 iccpartiltu 45621 itscnhlinecirc02p 46878 |
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