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Theorem relrngo 25496
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 25495 . 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 5040 1  |-  Rel  RingOps
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733    X. cxp 4911   ran crn 4914   Rel wrel 4918   -->wf 5492  (class class class)co 6196   AbelOpcablo 25400   RingOpscrngo 25494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-opab 4426  df-xp 4919  df-rel 4920  df-rngo 25495
This theorem is referenced by:  isrngo  25497  rngoi  25499  rngoablo2  25541  rngosn3  25545  isdrngo1  30525  iscrngo2  30561
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