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Theorem relrngo 24011
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 24010 . 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 5068 1  |-  Rel  RingOps
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797    X. cxp 4941   ran crn 4944   Rel wrel 4948   -->wf 5517  (class class class)co 6195   AbelOpcablo 23915   RingOpscrngo 24009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4454  df-xp 4949  df-rel 4950  df-rngo 24010
This theorem is referenced by:  isrngo  24012  rngoi  24014  rngoablo2  24056  rngosn3  24060  isdrngo1  28905  iscrngo2  28941
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