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Theorem relrngo 21965
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 21964 . 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 5000 1  |-  Rel  RingOps
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    X. cxp 4876   ran crn 4879   Rel wrel 4883   -->wf 5450  (class class class)co 6081   AbelOpcablo 21869   RingOpscrngo 21963
This theorem is referenced by:  isrngo  21966  rngoi  21968  rngoablo2  22010  rngosn3  22014  isdrngo1  26572  iscrngo2  26608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885  df-rngo 21964
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