Theorem irrednu 20341

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	⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈)

Proof of Theorem irrednu

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		irrednu.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2730	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20337	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
6	5	simp2bi 1146	1 ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228  Unitcui 20271  Irredcir 20272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-irred 20275

This theorem is referenced by:  irredn1  20342  mxidlirred  33450  rprmirredb  33510  irredminply  33713