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Theorem pnfnlt 11362
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 9652 . . . . . . 7  |- +oo  e/  RR
21neli 2792 . . . . . 6  |-  -. +oo  e.  RR
32intnanr 915 . . . . 5  |-  -.  ( +oo  e.  RR  /\  A  e.  RR )
43intnanr 915 . . . 4  |-  -.  (
( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )
5 pnfnemnf 11351 . . . . . 6  |- +oo  =/= -oo
65neii 2656 . . . . 5  |-  -. +oo  = -oo
76intnanr 915 . . . 4  |-  -.  ( +oo  = -oo  /\  A  = +oo )
84, 7pm3.2ni 854 . . 3  |-  -.  (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )
92intnanr 915 . . . 4  |-  -.  ( +oo  e.  RR  /\  A  = +oo )
106intnanr 915 . . . 4  |-  -.  ( +oo  = -oo  /\  A  e.  RR )
119, 10pm3.2ni 854 . . 3  |-  -.  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 854 . 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 11346 . . 3  |- +oo  e.  RR*
14 ltxr 11349 . . 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 1395    e. wcel 1819   class class class wbr 4456   RRcr 9508    <RR cltrr 9513   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650
This theorem is referenced by:  pnfge  11364  xrltnsym  11368  xrlttr  11371  qbtwnxr  11424  xltnegi  11440  xmullem2  11482  xrinfmexpnf  11522  xrsupsslem  11523  xrinfmsslem  11524  xrub  11528  supxrpnf  11535  supxrunb1  11536  supxrunb2  11537  xrinfm0  11553  lt6abl  17023  pnfnei  19847  metdstri  21480  esumpcvgval  28240
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