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Theorem enrex 9433
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 9353 . . . 4  |-  P.  e.  _V
21, 1xpex 6577 . . 3  |-  ( P. 
X.  P. )  e.  _V
32, 2xpex 6577 . 2  |-  ( ( P.  X.  P. )  X.  ( P.  X.  P. ) )  e.  _V
4 df-enr 9422 . . 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 5063 . . 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 3519 . 2  |-  ~R  C_  (
( P.  X.  P. )  X.  ( P.  X.  P. ) )
73, 6ssexi 4582 1  |-  ~R  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106   <.cop 4022   {copab 4496    X. cxp 4986  (class class class)co 6270   P.cnp 9226    +P. cpp 9228    ~R cer 9231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-om 6674  df-ni 9239  df-nq 9279  df-np 9348  df-enr 9422
This theorem is referenced by:  addsrpr  9441  mulsrpr  9442  ltsrpr  9443  0r  9446  1sr  9447  m1r  9448  addclsr  9449  mulclsr  9450  recexsrlem  9469
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