Theorem isidom 19125

Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)

Assertion

Ref	Expression
isidom	⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Proof of Theorem isidom

Step	Hyp	Ref	Expression
1		df-idom 19106	. 2 ⊢ IDomn = (CRing ∩ Domn)
2	1	elin2 3763	1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  CRingccrg 18371  Domncdomn 19101  IDomncidom 19102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-idom 19106

This theorem is referenced by:  fldidom  19126  fiidomfld  19129  znfld  19728  znidomb  19729  recvs  22754  ply1idom  23688  fta1glem1  23729  fta1glem2  23730  fta1g  23731  fta1b  23733  lgsqrlem1  24871  lgsqrlem2  24872  lgsqrlem3  24873  lgsqrlem4  24874  idomrootle  36792  idomodle  36793  proot1mul  36796  proot1hash  36797