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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 2610 | . . . 4 ⊢ +∞ = +∞ | |
2 | orc 399 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) | |
3 | 1, 2 | mpan2 703 | . . 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 9964 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
6 | pnfxr 9971 | . . 3 ⊢ +∞ ∈ ℝ* | |
7 | ltxr 11825 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancl 693 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) |
9 | 4, 8 | mpbird 246 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 <ℝ cltrr 9819 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-pnf 9955 df-xr 9957 df-ltxr 9958 |
This theorem is referenced by: ltpnfd 11831 0ltpnf 11832 xrlttri 11848 xrlttr 11849 xrrebnd 11873 xrre 11874 qbtwnxr 11905 xltnegi 11921 xrinfmsslem 12010 xrub 12014 supxrunb1 12021 supxrunb2 12022 elioc2 12107 elicc2 12109 ioomax 12119 ioopos 12121 elioopnf 12138 elicopnf 12140 difreicc 12175 hashbnd 12985 hashnnn0genn0 12993 hashv01gt1 12995 fprodge0 14563 fprodge1 14565 pcadd 15431 ramubcl 15560 rge0srg 19636 mnfnei 20835 xblss2ps 22016 icopnfcld 22381 iocmnfcld 22382 blcvx 22409 xrtgioo 22417 reconnlem1 22437 xrge0tsms 22445 iccpnfhmeo 22552 ioombl1lem4 23136 icombl1 23138 uniioombllem1 23155 mbfmax 23222 ismbf3d 23227 itg2seq 23315 lhop2 23582 dvfsumlem2 23594 logccv 24209 xrlimcnp 24495 pntleme 25097 upgrfi 25758 umgrafi 25851 frgrawopreglem2 26572 topnfbey 26717 isblo3i 27040 htthlem 27158 xlt2addrd 28913 dfrp2 28922 fsumrp0cl 29026 pnfinf 29068 xrge0tsmsd 29116 xrge0slmod 29175 xrge0iifcnv 29307 xrge0iifiso 29309 xrge0iifhom 29311 lmxrge0 29326 esumcst 29452 esumcvgre 29480 voliune 29619 volfiniune 29620 sxbrsigalem0 29660 orvcgteel 29856 dstfrvclim1 29866 itg2addnclem2 32632 asindmre 32665 dvasin 32666 dvacos 32667 rfcnpre3 38215 supxrgere 38490 supxrgelem 38494 xrlexaddrp 38509 infxr 38524 limsupre 38708 icccncfext 38773 fourierdlem111 39110 fourierdlem113 39112 fouriersw 39124 sge0iunmptlemre 39308 sge0rpcpnf 39314 sge0xaddlem1 39326 meaiuninclem 39373 hoidmvlelem5 39489 ovolval5lem1 39542 pimltpnf 39593 iccpartiltu 39960 |
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