Theorem flddivrng 38708

Description: Obsolete theorem, use flddrngd 20891 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 38701	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4189	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 3984	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3934	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∩ cin 3905  DivRingOpscdrng 38657  Com2ccm2 38698  Fldcfld 38700

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3913  df-ss 3923  df-fld 38701

This theorem is used by:  isfld2  38714  isfldidl  38777