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Theorem irrednu 15815
 Description: An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i Irred
irrednu.u Unit
Assertion
Ref Expression
irrednu

Proof of Theorem irrednu
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3
2 irrednu.u . . 3 Unit
3 irredn0.i . . 3 Irred
4 eqid 2438 . . 3
51, 2, 3, 4isirred2 15811 . 2
65simp2bi 974 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 359   wceq 1653   wcel 1726  wral 2707  cfv 5457  (class class class)co 6084  cbs 13474  cmulr 13535  Unitcui 15749  Irredcir 15750 This theorem is referenced by:  irredn1  15816 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-irred 15753
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