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Theorem isnirred 15805
 Description: The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irred.1
irred.2 Unit
irred.3 Irred
irred.4
irred.5
Assertion
Ref Expression
isnirred
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem isnirred
StepHypRef Expression
1 irred.4 . . . . . . 7
21eleq2i 2500 . . . . . 6
3 eldif 3330 . . . . . 6
42, 3bitri 241 . . . . 5
54baibr 873 . . . 4
6 df-ne 2601 . . . . . . . . 9
76ralbii 2729 . . . . . . . 8
8 ralnex 2715 . . . . . . . 8
97, 8bitri 241 . . . . . . 7
109ralbii 2729 . . . . . 6
11 ralnex 2715 . . . . . 6
1210, 11bitr2i 242 . . . . 5
1312a1i 11 . . . 4
145, 13anbi12d 692 . . 3
15 ioran 477 . . 3
16 irred.1 . . . 4
17 irred.2 . . . 4 Unit
18 irred.3 . . . 4 Irred
19 irred.5 . . . 4
2016, 17, 18, 1, 19isirred 15804 . . 3
2114, 15, 203bitr4g 280 . 2
2221con1bid 321 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725   wne 2599  wral 2705  wrex 2706   cdif 3317  cfv 5454  (class class class)co 6081  cbs 13469  cmulr 13530  Unitcui 15744  Irredcir 15745 This theorem is referenced by:  irredn0  15808  irredrmul  15812 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-irred 15748
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