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Theorem relrngo 21044
Description: The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
relrngo  |-  Rel  RingOps

Proof of Theorem relrngo
Dummy variables  g  h  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngo 21043 . 2  |-  RingOps  =  { <. g ,  h >.  |  ( ( g  e. 
AbelOp  /\  h : ( ran  g  X.  ran  g ) --> ran  g
)  /\  ( A. x  e.  ran  g A. y  e.  ran  g A. z  e.  ran  g ( ( ( x h y ) h z )  =  ( x h ( y h z ) )  /\  ( x h ( y g z ) )  =  ( ( x h y ) g ( x h z ) )  /\  ( ( x g y ) h z )  =  ( ( x h z ) g ( y h z ) ) )  /\  E. x  e. 
ran  g A. y  e.  ran  g ( ( x h y )  =  y  /\  (
y h x )  =  y ) ) ) }
21relopabi 4811 1  |-  Rel  RingOps
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   Rel wrel 4694   -->wf 5251  (class class class)co 5858   AbelOpcablo 20948   RingOpscrngo 21042
This theorem is referenced by:  isrngo  21045  rngoi  21047  rngoablo2  21089  rngosn3  21093  isdrngo1  26587  iscrngo2  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-rngo 21043
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