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Theorem irrednu 20403
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 2726 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 irrednu.u . . 3 𝑈 = (Unit‘𝑅)
3 irredn0.i . . 3 𝐼 = (Irred‘𝑅)
4 eqid 2726 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isirred2 20399 . 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 845   = wceq 1534  wcel 2099  wral 3051  cfv 6546  (class class class)co 7416  Basecbs 17208  .rcmulr 17262  Unitcui 20333  Irredcir 20334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-irred 20337
This theorem is referenced by:  irredn1  20404  mxidlirred  33353  rprmirredb  33413  irredminply  33589
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