Theorem flddivrng 33798

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

Step	Hyp	Ref	Expression
1		df-fld 33791	. . 3 |- Fld = ( DivRingOps i^i Com2 )
2		inss1 3833	. . 3 |- ( DivRingOps i^i Com2 ) C_ DivRingOps
3	1, 2	eqsstri 3635	. 2 |- Fld C_ DivRingOps
4	3	sseli 3599	1 |- ( K e. Fld -> K e. DivRingOps )

Syntax hints:   -> wi 4   e. wcel 1990   i^i cin 3573   DivRingOpscdrng 33747   Com2ccm2 33788   Fldcfld 33790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-fld 33791

This theorem is referenced by:  isfld2  33804  isfldidl  33867