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Theorem nndceq0 4282
Description: A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
Assertion
Ref Expression
nndceq0  om DECID  (/)

Proof of Theorem nndceq0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . 4  (/)  (/)  (/)  (/)
21notbid 591 . . . 4  (/)  (/)  (/)  (/)
31, 2orbi12d 706 . . 3  (/)  (/)  (/)  (/)  (/)  (/)  (/)
4 eqeq1 2043 . . . 4  (/)  (/)
54notbid 591 . . . 4  (/)  (/)
64, 5orbi12d 706 . . 3  (/)  (/)  (/)  (/)
7 eqeq1 2043 . . . 4  suc  (/)  suc  (/)
87notbid 591 . . . 4  suc  (/) 
suc  (/)
97, 8orbi12d 706 . . 3  suc  (/)  (/)  suc  (/)  suc  (/)
10 eqeq1 2043 . . . 4  (/)  (/)
1110notbid 591 . . . 4  (/)  (/)
1210, 11orbi12d 706 . . 3  (/)  (/)  (/)  (/)
13 eqid 2037 . . . 4  (/)  (/)
1413orci 649 . . 3  (/)  (/)  (/)  (/)
15 peano3 4262 . . . . . 6  om  suc  =/=  (/)
1615neneqd 2221 . . . . 5  om  suc  (/)
1716olcd 652 . . . 4  om  suc  (/)  suc  (/)
1817a1d 22 . . 3  om  (/)  (/)  suc  (/)  suc  (/)
193, 6, 9, 12, 14, 18finds 4266 . 2  om  (/)  (/)
20 df-dc 742 . 2 DECID  (/)  (/)  (/)
2119, 20sylibr 137 1  om DECID  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wo 628  DECID wdc 741   wceq 1242   wcel 1390   (/)c0 3218   suc csuc 4068   omcom 4256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257
This theorem is referenced by:  elni2  6298  indpi  6326
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