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Theorem pnfnlt 10730
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 9132 . . . . . . 7  |-  +oo  e/  RR
21neli 2699 . . . . . 6  |-  -.  +oo  e.  RR
32intnanr 883 . . . . 5  |-  -.  (  +oo  e.  RR  /\  A  e.  RR )
43intnanr 883 . . . 4  |-  -.  (
(  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )
5 pnfnemnf 10722 . . . . . 6  |-  +oo  =/=  -oo
65neii 2605 . . . . 5  |-  -.  +oo  =  -oo
76intnanr 883 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  =  +oo )
84, 7pm3.2ni 829 . . 3  |-  -.  (
( (  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )  \/  (  +oo  =  -oo  /\  A  =  +oo ) )
92intnanr 883 . . . 4  |-  -.  (  +oo  e.  RR  /\  A  =  +oo )
106intnanr 883 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  e.  RR )
119, 10pm3.2ni 829 . . 3  |-  -.  (
(  +oo  e.  RR  /\  A  =  +oo )  \/  (  +oo  =  -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 829 . 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 10718 . . 3  |-  +oo  e.  RR*
14 ltxr 10720 . . 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 653 . 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 296 1  |-  ( A  e.  RR*  ->  -.  +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   RRcr 8994    <RR cltrr 8999    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124    < clt 9125
This theorem is referenced by:  pnfge  10732  xrltnsym  10735  xrlttr  10738  qbtwnxr  10791  xltnegi  10807  xmullem2  10849  xrinfmexpnf  10889  xrsupsslem  10890  xrinfmsslem  10891  xrub  10895  supxrpnf  10902  supxrunb1  10903  supxrunb2  10904  xrinfm0  10920  lt6abl  15509  pnfnei  17289  metdstri  18886  esumpcvgval  24473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130
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