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Theorem flddivrng 25543
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 25542 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
2 inss1 3714 . . 3  |-  ( DivRingOps  i^i  Com2 )  C_  DivRingOps
31, 2eqsstri 3529 . 2  |-  Fld  C_  DivRingOps
43sseli 3495 1  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    i^i cin 3470   DivRingOpscdrng 25533   Com2ccm2 25538   Fldcfld 25541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-fld 25542
This theorem is referenced by:  isfld2  30564  isfldidl  30627
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