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Theorem prmidlssidl 32265
Description: Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
Assertion
Ref Expression
prmidlssidl (𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) βŠ† (LIdealβ€˜π‘…))

Proof of Theorem prmidlssidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 prmidlidl 32264 . . 3 ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑖 ∈ (LIdealβ€˜π‘…))
21ex 414 . 2 (𝑅 ∈ Ring β†’ (𝑖 ∈ (PrmIdealβ€˜π‘…) β†’ 𝑖 ∈ (LIdealβ€˜π‘…)))
32ssrdv 3951 1 (𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) βŠ† (LIdealβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107   βŠ† wss 3911  β€˜cfv 6497  Ringcrg 19969  LIdealclidl 20647  PrmIdealcprmidl 32255
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-prmidl 32256
This theorem is referenced by:  0ringprmidl  32270  rspecbas  32503  rspectopn  32505
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