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Theorem pnfnlt 10556
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 8961 . . . . . . 7  |-  +oo  e/  RR
2 df-nel 2524 . . . . . . 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 10548 . . . . . 6  |-  +oo  =/=  -oo
7 df-ne 2523 . . . . . 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 10544 . . 3  |-  +oo  e.  RR*
16 ltxr 10546 . . 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 1642    e. wcel 1710    =/= wne 2521    e/ wnel 2522   class class class wbr 4102   RRcr 8823    <RR cltrr 8828    +oocpnf 8951    -oocmnf 8952   RR*cxr 8953    < clt 8954
This theorem is referenced by:  pnfge  10558  xrltnsym  10560  xrlttr  10563  qbtwnxr  10616  xltnegi  10632  xmullem2  10674  xrinfmexpnf  10713  xrsupsslem  10714  xrinfmsslem  10715  xrub  10719  supxrpnf  10726  supxrunb1  10727  supxrunb2  10728  xrinfm0  10744  lt6abl  15274  pnfnei  17050  metdstri  18452  esumpcvgval  23734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-xp 4774  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959
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