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Theorem pnfnlt 10467
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 8874 . . . . . . 7  |-  +oo  e/  RR
2 df-nel 2449 . . . . . . 7  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
31, 2mpbi 199 . . . . . 6  |-  -.  +oo  e.  RR
43intnanr 881 . . . . 5  |-  -.  (  +oo  e.  RR  /\  A  e.  RR )
54intnanr 881 . . . 4  |-  -.  (
(  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )
6 pnfnemnf 10459 . . . . . 6  |-  +oo  =/=  -oo
7 df-ne 2448 . . . . . 6  |-  (  +oo  =/=  -oo  <->  -.  +oo  =  -oo )
86, 7mpbi 199 . . . . 5  |-  -.  +oo  =  -oo
98intnanr 881 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  =  +oo )
105, 9pm3.2ni 827 . . 3  |-  -.  (
( (  +oo  e.  RR  /\  A  e.  RR )  /\  +oo  <RR  A )  \/  (  +oo  =  -oo  /\  A  =  +oo ) )
113intnanr 881 . . . 4  |-  -.  (  +oo  e.  RR  /\  A  =  +oo )
128intnanr 881 . . . 4  |-  -.  (  +oo  =  -oo  /\  A  e.  RR )
1311, 12pm3.2ni 827 . . 3  |-  -.  (
(  +oo  e.  RR  /\  A  =  +oo )  \/  (  +oo  =  -oo  /\  A  e.  RR ) )
1410, 13pm3.2ni 827 . 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 10455 . . 3  |-  +oo  e.  RR*
16 ltxr 10457 . . 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 651 . 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 294 1  |-  ( A  e.  RR*  ->  -.  +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   class class class wbr 4023   RRcr 8736    <RR cltrr 8741    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867
This theorem is referenced by:  pnfge  10469  xrltnsym  10471  xrlttr  10474  qbtwnxr  10527  xltnegi  10543  xmullem2  10585  xrinfmexpnf  10624  xrsupsslem  10625  xrinfmsslem  10626  xrub  10630  supxrpnf  10637  supxrunb1  10638  supxrunb2  10639  xrinfm0  10655  lt6abl  15181  pnfnei  16950  metdstri  18355  esumpcvgval  23446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872
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