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Theorem mnfltpnf 11360
Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
mnfltpnf  |- -oo  < +oo

Proof of Theorem mnfltpnf
StepHypRef Expression
1 eqid 2457 . . . 4  |- -oo  = -oo
2 eqid 2457 . . . 4  |- +oo  = +oo
3 olc 384 . . . 4  |-  ( ( -oo  = -oo  /\ +oo  = +oo )  -> 
( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo  <RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) ) )
41, 2, 3mp2an 672 . . 3  |-  ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )
54orci 390 . 2  |-  ( ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo  <RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo )
)  \/  ( ( -oo  e.  RR  /\ +oo  = +oo )  \/  ( -oo  = -oo  /\ +oo  e.  RR ) ) )
6 mnfxr 11348 . . 3  |- -oo  e.  RR*
7 pnfxr 11346 . . 3  |- +oo  e.  RR*
8 ltxr 11349 . . 3  |-  ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  ->  ( -oo  < +oo  <->  ( ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )  \/  ( ( -oo  e.  RR  /\ +oo  = +oo )  \/  ( -oo  = -oo  /\ +oo  e.  RR ) ) ) ) )
96, 7, 8mp2an 672 . 2  |-  ( -oo  < +oo  <->  ( ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )  \/  ( ( -oo  e.  RR  /\ +oo  = +oo )  \/  ( -oo  = -oo  /\ +oo  e.  RR ) ) ) )
105, 9mpbir 209 1  |- -oo  < +oo
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   RRcr 9508    <RR cltrr 9513   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650
This theorem is referenced by:  mnfltxr  11361  xrlttri  11370  xrlttr  11371  xltnegi  11440
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