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

Theorem ltrelsr 7284
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
Assertion
Ref Expression
ltrelsr  |-  <R  C_  ( R.  X.  R. )

Proof of Theorem ltrelsr
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltr 7276 . 2  |-  <R  =  { <. x ,  y
>.  |  ( (
x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
2 opabssxp 4512 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }  C_  ( R.  X.  R. )
31, 2eqsstri 3056 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2999   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436  (class class class)co 5652   [cec 6290    +P. cpp 6852    <P cltp 6854    ~R cer 6855   R.cnr 6856    <R cltr 6862
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-opab 3900  df-xp 4444  df-ltr 7276
This theorem is referenced by:  gt0srpr  7294  recexgt0sr  7319  addgt0sr  7321  mulgt0sr  7323  caucvgsrlemcl  7334  caucvgsrlemasr  7335  caucvgsrlemfv  7336  ltresr  7376  axpre-ltirr  7417  axpre-lttrn  7419
  Copyright terms: Public domain W3C validator