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Theorem mnfltpnf 10465
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 2283 . . . 4  |-  -oo  =  -oo
2 eqid 2283 . . . 4  |-  +oo  =  +oo
3 olc 373 . . . 4  |-  ( ( 
-oo  =  -oo  /\  +oo  =  +oo )  -> 
( ( (  -oo  e.  RR  /\  +oo  e.  RR )  /\  -oo  <RR  +oo )  \/  (  -oo  =  -oo  /\ 
+oo  =  +oo )
) )
41, 2, 3mp2an 653 . . 3  |-  ( ( (  -oo  e.  RR  /\ 
+oo  e.  RR )  /\  -oo  <RR  +oo )  \/  (  -oo  =  -oo  /\  +oo  =  +oo ) )
54orci 379 . 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 10456 . . 3  |-  -oo  e.  RR*
7 pnfxr 10455 . . 3  |-  +oo  e.  RR*
8 ltxr 10457 . . 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 653 . 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 200 1  |-  -oo  <  +oo
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   RRcr 8736    <RR cltrr 8741    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867
This theorem is referenced by:  mnfltxr  10466  xrlttri  10473  xrlttr  10474  xltnegi  10543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872
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