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Theorem flddivrng 21098
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 21097 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3402 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3221 . 2  |-  Fld  C_  DivRingOps
43sseli 3189 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164   DivRingOpscdrng 21088   Com2ccm2 21093   Fldcfld 21096
This theorem is referenced by:  zrfld  25538  isfld2  26733  isfldidl  26796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-fld 21097
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