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Theorem flddivrng 21993
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 21992 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3553 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3370 . 2  |-  Fld  C_  DivRingOps
43sseli 3336 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    i^i cin 3311   DivRingOpscdrng 21983   Com2ccm2 21988   Fldcfld 21991
This theorem is referenced by:  isfld2  26569  isfldidl  26632
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-fld 21992
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