ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mnflt Unicode version

Theorem mnflt 9251
Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
mnflt  |-  ( A  e.  RR  -> -oo  <  A )

Proof of Theorem mnflt
StepHypRef Expression
1 eqid 2088 . . . 4  |- -oo  = -oo
2 olc 667 . . . 4  |-  ( ( -oo  = -oo  /\  A  e.  RR )  ->  ( ( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) )
31, 2mpan 415 . . 3  |-  ( A  e.  RR  ->  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) )
43olcd 688 . 2  |-  ( A  e.  RR  ->  (
( ( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  ( ( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) )
5 mnfxr 7542 . . 3  |- -oo  e.  RR*
6 rexr 7531 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
7 ltxr 9244 . . 3  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <  A  <->  ( (
( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) ) )
85, 6, 7sylancr 405 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  <->  ( (
( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) ) )
94, 8mpbird 165 1  |-  ( A  e.  RR  -> -oo  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   class class class wbr 3845   RRcr 7347    <RR cltrr 7352   +oocpnf 7517   -oocmnf 7518   RR*cxr 7519    < clt 7520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-cnex 7434
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525
This theorem is referenced by:  mnflt0  9252  mnfltxr  9254  xrlttr  9263  xrltso  9264  xrlttri3  9265  ngtmnft  9278  xrrebnd  9279  xrre3  9282  xltnegi  9295  elico2  9353  elicc2  9354  ioomax  9364  elioomnf  9384  qbtwnxr  9665
  Copyright terms: Public domain W3C validator