Theorem prmidlssidl 21606

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 21605	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅))
2	1	ex 418	. 2 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (PrmIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅)))
3	2	ssrdv 3937	1 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3899  ‘cfv 6531  Ringcrg 20439  LIdealclidl 21464  PrmIdealcprmidl 21596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-prmidl 21597

This theorem is used by:  0ringprmidl  21613  rspecbas  34479  rspectopn  34481