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Theorem mnfltpnf 11216
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 2454 . . . 4  |- -oo  = -oo
2 eqid 2454 . . . 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 11204 . . 3  |- -oo  e.  RR*
7 pnfxr 11202 . . 3  |- +oo  e.  RR*
8 ltxr 11205 . . 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 1370    e. wcel 1758   class class class wbr 4399   RRcr 9391    <RR cltrr 9396   +oocpnf 9525   -oocmnf 9526   RR*cxr 9527    < clt 9528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-xp 4953  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533
This theorem is referenced by:  mnfltxr  11217  xrlttri  11226  xrlttr  11227  xltnegi  11296
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