Theorem flddivrng 16113

Description: A field is a division ring.

Assertion

Ref	Expression
flddivrng	|- (K e. Fld -> K e. DivRing)

Proof of Theorem flddivrng

Step	Hyp	Ref	Expression
1		df-fld 10398	. . 3 |- Fld = (DivRing i^i Com2)
2	1	eleq2i 1961	. 2 |- (K e. Fld <-> K e. (DivRing i^i Com2))
3		inss1 2812	. . 3 |- (DivRing i^i Com2) C_ DivRing
4	3	sseli 2617	. 2 |- (K e. (DivRing i^i Com2) -> K e. DivRing)
5	2, 4	sylbi 216	1 |- (K e. Fld -> K e. DivRing)

Syntax hints:   -> wi 3   e. wcel 1300   i^i cin 2592  DivRingcdrng 9491  Com2ccm2 10394  Fldcfld 10397

This theorem is referenced by:  isfld2 16153  isfldidl 16216

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-fld 10398

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