Theorem isidom 20922

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 20901	. 2 ⊢ IDomn = (CRing ∩ Domn)
2	1	elin2 4198	1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  CRingccrg 20057  Domncdomn 20896  IDomncidom 20897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-idom 20901

This theorem is referenced by:  fldidom  20923  fldidomOLD  20924  fiidomfld  20927  znfld  21116  znidomb  21117  recvsOLD  24663  ply1idom  25642  fta1glem1  25683  fta1glem2  25684  fta1g  25685  fta1b  25687  lgsqrlem1  26849  lgsqrlem2  26850  lgsqrlem3  26851  lgsqrlem4  26852  dvdsruasso  32490  qsidomlem1  32571  qsidomlem2  32572  idomrootle  41937  idomodle  41938  proot1mul  41941  proot1hash  41942