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Theorem enrex 8708
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 8626 . . . 4  |-  P.  e.  _V
21, 1xpex 4817 . . 3  |-  ( P. 
X.  P. )  e.  _V
32, 2xpex 4817 . 2  |-  ( ( P.  X.  P. )  X.  ( P.  X.  P. ) )  e.  _V
4 df-enr 8697 . . 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 4778 . . 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 3221 . 2  |-  ~R  C_  (
( P.  X.  P. )  X.  ( P.  X.  P. ) )
73, 6ssexi 4175 1  |-  ~R  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   {copab 4092    X. cxp 4703  (class class class)co 5874   P.cnp 8497    +P. cpp 8499    ~R cer 8504
This theorem is referenced by:  addsrpr  8713  mulsrpr  8714  ltsrpr  8715  0r  8718  1sr  8719  m1r  8720  addclsr  8721  mulclsr  8722  recexsrlem  8741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-ni 8512  df-nq 8552  df-np 8621  df-enr 8697
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