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Theorem mnfltpnf 9798
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 2187 . . . 4  |- -oo  = -oo
2 eqid 2187 . . . 4  |- +oo  = +oo
3 olc 712 . . . 4  |-  ( ( -oo  = -oo  /\ +oo  = +oo )  -> 
( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo  <RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) ) )
41, 2, 3mp2an 426 . . 3  |-  ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )
54orci 732 . 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 8027 . . 3  |- -oo  e.  RR*
7 pnfxr 8023 . . 3  |- +oo  e.  RR*
8 ltxr 9788 . . 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 426 . 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 146 1  |- -oo  < +oo
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1363    e. wcel 2158   class class class wbr 4015   RRcr 7823    <RR cltrr 7828   +oocpnf 8002   -oocmnf 8003   RR*cxr 8004    < clt 8005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010
This theorem is referenced by:  mnfltxr  9799  xrlttr  9808  xrltso  9809  xrlttri3  9810  nltpnft  9827  npnflt  9828  ngtmnft  9830  nmnfgt  9831  xltnegi  9848  xposdif  9895
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