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Theorem flddivrng 25988
Description: A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
flddivrng  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )

Proof of Theorem flddivrng
StepHypRef Expression
1 df-fld 25987 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3688 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3500 . 2  |-  Fld  C_  DivRingOps
43sseli 3466 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870    i^i cin 3441   DivRingOpscdrng 25978   Com2ccm2 25983   Fldcfld 25986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-in 3449  df-ss 3456  df-fld 25987
This theorem is referenced by:  isfld2  31942  isfldidl  32005
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