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Theorem irrednu 19458
 Description: An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i 𝐼 = (Irred‘𝑅)
irrednu.u 𝑈 = (Unit‘𝑅)
Assertion
Ref Expression
irrednu (𝑋𝐼 → ¬ 𝑋𝑈)

Proof of Theorem irrednu
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 irrednu.u . . 3 𝑈 = (Unit‘𝑅)
3 irredn0.i . . 3 𝐼 = (Irred‘𝑅)
4 eqid 2824 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isirred2 19454 . 2 (𝑋𝐼 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ¬ 𝑋𝑈 ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
65simp2bi 1143 1 (𝑋𝐼 → ¬ 𝑋𝑈)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  .rcmulr 16566  Unitcui 19392  Irredcir 19393 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-irred 19396 This theorem is referenced by:  irredn1  19459
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