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Theorem mnfltpnf 9354
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 2095 . . . 4  |- -oo  = -oo
2 eqid 2095 . . . 4  |- +oo  = +oo
3 olc 670 . . . 4  |-  ( ( -oo  = -oo  /\ +oo  = +oo )  -> 
( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo  <RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) ) )
41, 2, 3mp2an 418 . . 3  |-  ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )
54orci 688 . 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 7641 . . 3  |- -oo  e.  RR*
7 pnfxr 7637 . . 3  |- +oo  e.  RR*
8 ltxr 9345 . . 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 418 . 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 145 1  |- -oo  < +oo
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 667    = wceq 1296    e. wcel 1445   class class class wbr 3867   RRcr 7446    <RR cltrr 7451   +oocpnf 7616   -oocmnf 7617   RR*cxr 7618    < clt 7619
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-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624
This theorem is referenced by:  mnfltxr  9355  xrlttr  9364  xrltso  9365  xrlttri3  9366  nltpnft  9380  npnflt  9381  ngtmnft  9383  nmnfgt  9384  xltnegi  9401  xposdif  9448
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