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Theorem pnfnlt 10685
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 9087 . . . . . . 7  |-  +oo  e/  RR
2 df-nel 2574 . . . . . . 7  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
31, 2mpbi 200 . . . . . 6  |-  -.  +oo  e.  RR
43intnanr 882 . . . . 5  |-  -.  (  +oo  e.  RR  /\  A  e.  RR )
54intnanr 882 . . . 4  |-  -.  (
(  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )
6 pnfnemnf 10677 . . . . . 6  |-  +oo  =/=  -oo
7 df-ne 2573 . . . . . 6  |-  (  +oo  =/=  -oo  <->  -.  +oo  =  -oo )
86, 7mpbi 200 . . . . 5  |-  -.  +oo  =  -oo
98intnanr 882 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  =  +oo )
105, 9pm3.2ni 828 . . 3  |-  -.  (
( (  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )  \/  (  +oo  =  -oo  /\  A  =  +oo ) )
113intnanr 882 . . . 4  |-  -.  (  +oo  e.  RR  /\  A  =  +oo )
128intnanr 882 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  e.  RR )
1311, 12pm3.2ni 828 . . 3  |-  -.  (
(  +oo  e.  RR  /\  A  =  +oo )  \/  (  +oo  =  -oo  /\  A  e.  RR ) )
1410, 13pm3.2ni 828 . 2  |-  -.  (
( ( (  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )  \/  (  +oo  =  -oo  /\  A  =  +oo ) )  \/  ( (  +oo  e.  RR  /\  A  =  +oo )  \/  (  +oo  =  -oo  /\  A  e.  RR ) ) )
15 pnfxr 10673 . . 3  |-  +oo  e.  RR*
16 ltxr 10675 . . 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 ) ) ) ) )
1715, 16mpan 652 . 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 ) ) ) ) )
1814, 17mtbiri 295 1  |-  ( A  e.  RR*  ->  -.  +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571    e/ wnel 2572   class class class wbr 4176   RRcr 8949    <RR cltrr 8954    +oocpnf 9077    -oocmnf 9078   RR*cxr 9079    < clt 9080
This theorem is referenced by:  pnfge  10687  xrltnsym  10690  xrlttr  10693  qbtwnxr  10746  xltnegi  10762  xmullem2  10804  xrinfmexpnf  10844  xrsupsslem  10845  xrinfmsslem  10846  xrub  10850  supxrpnf  10857  supxrunb1  10858  supxrunb2  10859  xrinfm0  10875  lt6abl  15463  pnfnei  17242  metdstri  18838  esumpcvgval  24425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085
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