![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2760 | . . . 4 ⊢ +∞ = +∞ | |
2 | orc 399 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) | |
3 | 1, 2 | mpan2 709 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) |
4 | 3 | olcd 407 | . 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))) |
5 | rexr 10277 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
6 | pnfxr 10284 | . . 3 ⊢ +∞ ∈ ℝ* | |
7 | ltxr 12142 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancl 697 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) |
9 | 4, 8 | mpbird 247 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ℝcr 10127 <ℝ cltrr 10132 +∞cpnf 10263 -∞cmnf 10264 ℝ*cxr 10265 < clt 10266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-xp 5272 df-pnf 10268 df-xr 10270 df-ltxr 10271 |
This theorem is referenced by: ltpnfd 12148 0ltpnf 12149 xrlttri 12165 xrlttr 12166 xrrebnd 12192 xrre 12193 qbtwnxr 12224 xltnegi 12240 xrinfmsslem 12331 xrub 12335 supxrunb1 12342 supxrunb2 12343 elioc2 12429 elicc2 12431 ioomax 12441 ioopos 12443 elioopnf 12460 elicopnf 12462 difreicc 12497 hashbnd 13317 hashnnn0genn0 13325 hashv01gt1 13327 fprodge0 14923 fprodge1 14925 pcadd 15795 ramubcl 15924 rge0srg 20019 mnfnei 21227 xblss2ps 22407 icopnfcld 22772 iocmnfcld 22773 blcvx 22802 xrtgioo 22810 reconnlem1 22830 xrge0tsms 22838 iccpnfhmeo 22945 ioombl1lem4 23529 icombl1 23531 uniioombllem1 23549 mbfmax 23615 ismbf3d 23620 itg2seq 23708 lhop2 23977 dvfsumlem2 23989 logccv 24608 xrlimcnp 24894 pntleme 25496 upgrfi 26185 topnfbey 27636 isblo3i 27965 htthlem 28083 xlt2addrd 29832 dfrp2 29841 fsumrp0cl 30004 pnfinf 30046 xrge0tsmsd 30094 xrge0slmod 30153 xrge0iifcnv 30288 xrge0iifiso 30290 xrge0iifhom 30292 lmxrge0 30307 esumcst 30434 esumcvgre 30462 voliune 30601 volfiniune 30602 sxbrsigalem0 30642 orvcgteel 30838 dstfrvclim1 30848 itg2addnclem2 33775 asindmre 33808 dvasin 33809 dvacos 33810 rfcnpre3 39691 supxrgere 40047 supxrgelem 40051 xrlexaddrp 40066 infxr 40081 xrpnf 40214 limsupre 40376 limsuppnfdlem 40436 limsuppnflem 40445 liminflelimsupuz 40520 icccncfext 40603 fourierdlem111 40937 fourierdlem113 40939 fouriersw 40951 sge0iunmptlemre 41135 sge0rpcpnf 41141 sge0xaddlem1 41153 meaiuninclem 41200 hoidmvlelem5 41319 ovolval5lem1 41372 pimltpnf 41422 iccpartiltu 41868 |
Copyright terms: Public domain | W3C validator |