Theorem flddivrng 39199

Description: Obsolete theorem, use flddrngd 21045 instead. A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
flddivrng	⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Proof of Theorem flddivrng

Step	Hyp	Ref	Expression
1		df-fld 39192	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4136	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 3959	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3909	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  DivRingOpscdrng 39148  Com2ccm2 39189  Fldcfld 39191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-fld 39192

This theorem is used by:  isfld2  39205  isfldidl  39268