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Theorem flddivrng 21082
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 21081 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3389 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3208 . 2  |-  Fld  C_  DivRingOps
43sseli 3176 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151   DivRingOpscdrng 21072   Com2ccm2 21077   Fldcfld 21080
This theorem is referenced by:  zrfld  25435  isfld2  26630  isfldidl  26693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-fld 21081
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