Theorem isfld 18966

Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)

Assertion

Ref	Expression
isfld	⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

Proof of Theorem isfld

Step	Hyp	Ref	Expression
1		df-field 18960	. 2 ⊢ Field = (DivRing ∩ CRing)
2	1	elin2 3952	1 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∈ wcel 2145  CRingccrg 18756  DivRingcdr 18957  Fieldcfield 18958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-field 18960

This theorem is referenced by:  fldpropd  18985  rng1nfld  19493  fldidom  19520  fiidomfld  19523  refld  20182  recrng  20184  frlmphllem  20336  frlmphl  20337  rrxcph  23399  ply1pid  24159  lgseisenlem3  25323  lgseisenlem4  25324  ofldlt1  30153  ofldchr  30154  subofld  30156  isarchiofld  30157  reofld  30180  rearchi  30182  qqhrhm  30373  matunitlindflem1  33737  matunitlindflem2  33738  matunitlindf  33739  fldcat  42605  fldcatALTV  42623