Theorem idomdomd 32369

Description: An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypothesis

Ref	Expression
idomringd.1	⊢ (𝜑 → 𝑅 ∈ IDomn)

Assertion

Ref	Expression
idomdomd	⊢ (𝜑 → 𝑅 ∈ Domn)

Proof of Theorem idomdomd

Step	Hyp	Ref	Expression
1		idomringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn)
2		df-idom 20900	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2843	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4199	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3947  CRingccrg 20056  Domncdomn 20895  IDomncidom 20896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-idom 20900

This theorem is referenced by:  idomrcan  32372  minplyirredlem  32764  minplyirred  32765