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Theorem pnfnlt 9189
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 7473 . . . . . . 7  |- +oo  e/  RR
21neli 2348 . . . . . 6  |-  -. +oo  e.  RR
32intnanr 875 . . . . 5  |-  -.  ( +oo  e.  RR  /\  A  e.  RR )
43intnanr 875 . . . 4  |-  -.  (
( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )
5 pnfnemnf 7486 . . . . . 6  |- +oo  =/= -oo
65neii 2253 . . . . 5  |-  -. +oo  = -oo
76intnanr 875 . . . 4  |-  -.  ( +oo  = -oo  /\  A  = +oo )
84, 7pm3.2ni 760 . . 3  |-  -.  (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )
92intnanr 875 . . . 4  |-  -.  ( +oo  e.  RR  /\  A  = +oo )
106intnanr 875 . . . 4  |-  -.  ( +oo  = -oo  /\  A  e.  RR )
119, 10pm3.2ni 760 . . 3  |-  -.  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 760 . 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 7484 . . 3  |- +oo  e.  RR*
14 ltxr 9178 . . 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 415 . 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 633 1  |-  ( A  e.  RR*  ->  -. +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287    e. wcel 1436   class class class wbr 3820   RRcr 7293    <RR cltrr 7298   +oocpnf 7463   -oocmnf 7464   RR*cxr 7465    < clt 7466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-cnex 7380  ax-resscn 7381
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-xp 4417  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471
This theorem is referenced by:  pnfge  9191  xrltnsym  9195  xrlttr  9197  xrltso  9198  xltnegi  9229  qbtwnxr  9597
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