MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnfnlt Structured version   Unicode version

Theorem pnfnlt 11333
Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
pnfnlt  |-  ( A  e.  RR*  ->  -. +oo  <  A )

Proof of Theorem pnfnlt
StepHypRef Expression
1 pnfnre 9631 . . . . . . 7  |- +oo  e/  RR
21neli 2802 . . . . . 6  |-  -. +oo  e.  RR
32intnanr 913 . . . . 5  |-  -.  ( +oo  e.  RR  /\  A  e.  RR )
43intnanr 913 . . . 4  |-  -.  (
( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )
5 pnfnemnf 11322 . . . . . 6  |- +oo  =/= -oo
65neii 2666 . . . . 5  |-  -. +oo  = -oo
76intnanr 913 . . . 4  |-  -.  ( +oo  = -oo  /\  A  = +oo )
84, 7pm3.2ni 852 . . 3  |-  -.  (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )
92intnanr 913 . . . 4  |-  -.  ( +oo  e.  RR  /\  A  = +oo )
106intnanr 913 . . . 4  |-  -.  ( +oo  = -oo  /\  A  e.  RR )
119, 10pm3.2ni 852 . . 3  |-  -.  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 852 . 2  |-  -.  (
( ( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  ( ( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) )
13 pnfxr 11317 . . 3  |- +oo  e.  RR*
14 ltxr 11320 . . 3  |-  ( ( +oo  e.  RR*  /\  A  e.  RR* )  ->  ( +oo  <  A  <->  ( (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) ) ) )
1513, 14mpan 670 . 2  |-  ( A  e.  RR*  ->  ( +oo  <  A  <->  ( ( ( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) ) ) )
1612, 15mtbiri 303 1  |-  ( A  e.  RR*  ->  -. +oo  <  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   RRcr 9487    <RR cltrr 9492   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629
This theorem is referenced by:  pnfge  11335  xrltnsym  11339  xrlttr  11342  qbtwnxr  11395  xltnegi  11411  xmullem2  11453  xrinfmexpnf  11493  xrsupsslem  11494  xrinfmsslem  11495  xrub  11499  supxrpnf  11506  supxrunb1  11507  supxrunb2  11508  xrinfm0  11524  lt6abl  16688  pnfnei  19487  metdstri  21090  esumpcvgval  27724
  Copyright terms: Public domain W3C validator