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Theorem flddivrng 21853
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 21852 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3506 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3323 . 2  |-  Fld  C_  DivRingOps
43sseli 3289 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    i^i cin 3264   DivRingOpscdrng 21843   Com2ccm2 21848   Fldcfld 21851
This theorem is referenced by:  isfld2  26308  isfldidl  26371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-in 3272  df-ss 3279  df-fld 21852
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