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	⊢ (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 2740	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		irrednu.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2740	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20399	. 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 1152	1 ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  Unitcui 20333  Irredcir 20334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-irred 20337

This theorem is referenced by:  irredn1  20404  mxidlirred  33562  rprmirredb  33622  irredminply  33907