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Theorem prmidlssidl 31142
 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.
StepHypRef Expression
1 prmidlidl 31141 . . 3 ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅))
21ex 417 . 2 (𝑅 ∈ Ring → (𝑖 ∈ (PrmIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅)))
32ssrdv 3899 1 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ⊆ wss 3859  ‘cfv 6336  Ringcrg 19366  LIdealclidl 20011  PrmIdealcprmidl 31132 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7154  df-prmidl 31133 This theorem is referenced by:  0ringprmidl  31147  rspecbas  31337  rspectopn  31339
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