Theorem flddivrng 39042

Description: Obsolete theorem, use flddrngd 21003 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 39035	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4132	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 3955	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3905	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ∩ cin 4088  DivRingOpscdrng 38991  Com2ccm2 39032  Fldcfld 39034

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-in 4090  df-fld 39035

This theorem is used by:  isfld2  39048  isfldidl  39111