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Theorem enrex 9456
Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
Assertion
Ref Expression
enrex  |-  ~R  e.  _V

Proof of Theorem enrex
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 npex 9376 . . . 4  |-  P.  e.  _V
21, 1xpex 6599 . . 3  |-  ( P. 
X.  P. )  e.  _V
32, 2xpex 6599 . 2  |-  ( ( P.  X.  P. )  X.  ( P.  X.  P. ) )  e.  _V
4 df-enr 9445 . . 3  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
5 opabssxp 5080 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }  C_  (
( P.  X.  P. )  X.  ( P.  X.  P. ) )
64, 5eqsstri 3539 . 2  |-  ~R  C_  (
( P.  X.  P. )  X.  ( P.  X.  P. ) )
73, 6ssexi 4598 1  |-  ~R  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118   <.cop 4039   {copab 4510    X. cxp 5003  (class class class)co 6295   P.cnp 9249    +P. cpp 9251    ~R cer 9254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-om 6696  df-ni 9262  df-nq 9302  df-np 9371  df-enr 9445
This theorem is referenced by:  addsrpr  9464  mulsrpr  9465  ltsrpr  9466  0r  9469  1sr  9470  m1r  9471  addclsr  9472  mulclsr  9473  recexsrlem  9492
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