Theorem enrex 9888

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.

Step	Hyp	Ref	Expression
1		npex 9808	. . . 4 |- P. e. _V
2	1, 1	xpex 6962	. . 3 |- ( P. X. P. ) e. _V
3	2, 2	xpex 6962	. 2 |- ( ( P. X. P. ) X. ( P. X. P. ) ) e. _V
4		df-enr 9877	. . 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 5193	. . 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. ) )
6	4, 5	eqsstri 3635	. 2 |- ~R C_ ( ( P. X. P. ) X. ( P. X. P. ) )
7	3, 6	ssexi 4803	1 |- ~R e. _V

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   {copab 4712   X. cxp 5112  (class class class)co 6650   P.cnp 9681   +P. cpp 9683   ~R cer 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-ni 9694  df-nq 9734  df-np 9803  df-enr 9877

This theorem is referenced by:  addsrpr  9896  mulsrpr  9897  ltsrpr  9898  0r  9901  1sr  9902  m1r  9903  addclsr  9904  mulclsr  9905  recexsrlem  9924