Theorem prmidlssidl 33537

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 33536	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅))
2	1	ex 412	. 2 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (PrmIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅)))
3	2	ssrdv 3941	1 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3903  ‘cfv 6500  Ringcrg 20180  LIdealclidl 21173  PrmIdealcprmidl 33527

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-prmidl 33528

This theorem is referenced by:  0ringprmidl  33541  rspecbas  34042  rspectopn  34044