Theorem prmidlssidl 32558

Description: Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)

Assertion

Ref	Expression
prmidlssidl	⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))

Proof of Theorem prmidlssidl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmidlidl 32557	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅))
2	1	ex 413	. 2 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (PrmIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅)))
3	2	ssrdv 3988	1 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3948  ‘cfv 6543  Ringcrg 20055  LIdealclidl 20782  PrmIdealcprmidl 32548

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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-prmidl 32549

This theorem is referenced by:  0ringprmidl  32563  rspecbas  32840  rspectopn  32842