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Theorem mnfltpnf 9526
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 2117 . . . 4  |- -oo  = -oo
2 eqid 2117 . . . 4  |- +oo  = +oo
3 olc 685 . . . 4  |-  ( ( -oo  = -oo  /\ +oo  = +oo )  -> 
( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo  <RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) ) )
41, 2, 3mp2an 422 . . 3  |-  ( ( ( -oo  e.  RR  /\ +oo  e.  RR )  /\ -oo 
<RR +oo )  \/  ( -oo  = -oo  /\ +oo  = +oo ) )
54orci 705 . 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 7790 . . 3  |- -oo  e.  RR*
7 pnfxr 7786 . . 3  |- +oo  e.  RR*
8 ltxr 9517 . . 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 422 . 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 682    = wceq 1316    e. wcel 1465   class class class wbr 3899   RRcr 7587    <RR cltrr 7592   +oocpnf 7765   -oocmnf 7766   RR*cxr 7767    < clt 7768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773
This theorem is referenced by:  mnfltxr  9527  xrlttr  9536  xrltso  9537  xrlttri3  9538  nltpnft  9552  npnflt  9553  ngtmnft  9555  nmnfgt  9556  xltnegi  9573  xposdif  9620
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