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Theorem mnfltpnf 10481
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 2296 . . . 4  |-  -oo  =  -oo
2 eqid 2296 . . . 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 10472 . . 3  |-  -oo  e.  RR*
7 pnfxr 10471 . . 3  |-  +oo  e.  RR*
8 ltxr 10473 . . 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 1632    e. wcel 1696   class class class wbr 4039   RRcr 8752    <RR cltrr 8757    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883
This theorem is referenced by:  mnfltxr  10482  xrlttri  10489  xrlttr  10490  xltnegi  10559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888
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