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Theorem mnfltpnf 10723
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 2436 . . . 4  |-  -oo  =  -oo
2 eqid 2436 . . . 4  |-  +oo  =  +oo
3 olc 374 . . . 4  |-  ( ( 
-oo  =  -oo  /\  +oo  =  +oo )  -> 
( ( (  -oo  e.  RR  /\  +oo  e.  RR )  /\  -oo  <RR  +oo )  \/  (  -oo  =  -oo  /\ 
+oo  =  +oo )
) )
41, 2, 3mp2an 654 . . 3  |-  ( ( (  -oo  e.  RR  /\ 
+oo  e.  RR )  /\  -oo  <RR  +oo )  \/  (  -oo  =  -oo  /\  +oo  =  +oo ) )
54orci 380 . 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 10714 . . 3  |-  -oo  e.  RR*
7 pnfxr 10713 . . 3  |-  +oo  e.  RR*
8 ltxr 10715 . . 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 654 . 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 201 1  |-  -oo  <  +oo
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   RRcr 8989    <RR cltrr 8994    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119    < clt 9120
This theorem is referenced by:  mnfltxr  10724  xrlttri  10732  xrlttr  10733  xltnegi  10802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125
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