Metamath Proof Explorer

Theorem enrex

Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995) (New usage is discouraged.)

Ref Expression
Assertion enrex 
|- ~R e. _V

Proof

Step Hyp Ref Expression
1 npex 
 |-  P. e. _V
2 1 1 xpex 
 |-  ( P. X. P. ) e. _V
3 2 2 xpex 
 |-  ( ( P. X. P. ) X. ( P. X. P. ) ) e. _V
4 df-enr 
 |-  ~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 
 |-  { <. 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. ) )
6 4 5 eqsstri 
 |-  ~R C_ ( ( P. X. P. ) X. ( P. X. P. ) )
7 3 6 ssexi 
 |-  ~R e. _V