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 2731	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		irrednu.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2731	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  ._rcmulr 17159  Unitcui 20271  Irredcir 20272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-irred 20275

This theorem is referenced by:  irredn1  20342  mxidlirred  33432  rprmirredb  33492  irredminply  33724