ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltpnf Unicode version

Theorem ltpnf 8472
Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
ltpnf  RR  < +oo

Proof of Theorem ltpnf
StepHypRef Expression
1 eqid 2037 . . . 4 +oo +oo
2 orc 632 . . . 4  RR +oo +oo  RR +oo +oo -oo +oo  RR
31, 2mpan2 401 . . 3  RR  RR +oo +oo -oo +oo  RR
43olcd 652 . 2  RR  RR +oo  RR  <RR +oo -oo +oo +oo  RR +oo +oo -oo +oo  RR
5 rexr 6868 . . 3  RR  RR*
6 pnfxr 8462 . . 3 +oo  RR*
7 ltxr 8465 . . 3  RR* +oo  RR*  < +oo  RR +oo  RR  <RR +oo -oo +oo +oo  RR +oo +oo -oo +oo  RR
85, 6, 7sylancl 392 . 2  RR  < +oo  RR +oo  RR  <RR +oo -oo +oo +oo  RR +oo +oo -oo +oo  RR
94, 8mpbird 156 1  RR  < +oo
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390   class class class wbr 3755   RRcr 6710    <RR cltrr 6715   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856    < clt 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-cnex 6774
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  0ltpnf  8473  xrlttr  8486  xrltso  8487  xrlttri3  8488  nltpnft  8500  xrrebnd  8502  xrre  8503  xltnegi  8518  elioc2  8575  elicc2  8577  ioomax  8587  ioopos  8589  elioopnf  8606  elicopnf  8608
  Copyright terms: Public domain W3C validator