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Theorem pnfnlt 9466
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 7731 . . . . . . 7  |- +oo  e/  RR
21neli 2379 . . . . . 6  |-  -. +oo  e.  RR
32intnanr 898 . . . . 5  |-  -.  ( +oo  e.  RR  /\  A  e.  RR )
43intnanr 898 . . . 4  |-  -.  (
( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )
5 pnfnemnf 7744 . . . . . 6  |- +oo  =/= -oo
65neii 2284 . . . . 5  |-  -. +oo  = -oo
76intnanr 898 . . . 4  |-  -.  ( +oo  = -oo  /\  A  = +oo )
84, 7pm3.2ni 785 . . 3  |-  -.  (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )
92intnanr 898 . . . 4  |-  -.  ( +oo  e.  RR  /\  A  = +oo )
106intnanr 898 . . . 4  |-  -.  ( +oo  = -oo  /\  A  e.  RR )
119, 10pm3.2ni 785 . . 3  |-  -.  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 785 . 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 7742 . . 3  |- +oo  e.  RR*
14 ltxr 9455 . . 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 418 . 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 647 1  |-  ( A  e.  RR*  ->  -. +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   class class class wbr 3895   RRcr 7546    <RR cltrr 7551   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723    < clt 7724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-cnex 7636  ax-resscn 7637
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-xp 4505  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729
This theorem is referenced by:  pnfge  9468  xrltnsym  9472  xrlttr  9474  xrltso  9475  xltnegi  9511  xposdif  9558  qbtwnxr  9928  xrmaxiflemab  10908  xrmaxltsup  10919
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