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Theorem flddivrng 23924
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 23923 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3591 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3407 . 2  |-  Fld  C_  DivRingOps
43sseli 3373 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    i^i cin 3348   DivRingOpscdrng 23914   Com2ccm2 23919   Fldcfld 23922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2995  df-in 3356  df-ss 3363  df-fld 23923
This theorem is referenced by:  isfld2  28831  isfldidl  28894
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