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Theorem flddivrng 34285
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 ∈ Fld → 𝐾 ∈ DivRingOps)

Proof of Theorem flddivrng
StepHypRef Expression
1 df-fld 34278 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 4028 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 3831 . 2 Fld ⊆ DivRingOps
43sseli 3794 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  cin 3768  DivRingOpscdrng 34234  Com2ccm2 34275  Fldcfld 34277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-fld 34278
This theorem is referenced by:  isfld2  34291  isfldidl  34354
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