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Theorem nndceq0 4367
 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 DECID

Proof of Theorem nndceq0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . . 4
21notbid 602 . . . 4
31, 2orbi12d 717 . . 3
4 eqeq1 2062 . . . 4
54notbid 602 . . . 4
64, 5orbi12d 717 . . 3
7 eqeq1 2062 . . . 4
87notbid 602 . . . 4
97, 8orbi12d 717 . . 3
10 eqeq1 2062 . . . 4
1110notbid 602 . . . 4
1210, 11orbi12d 717 . . 3
13 eqid 2056 . . . 4
1413orci 660 . . 3
15 peano3 4347 . . . . . 6
1615neneqd 2241 . . . . 5
1716olcd 663 . . . 4
1817a1d 22 . . 3
193, 6, 9, 12, 14, 18finds 4351 . 2
20 df-dc 754 . 2 DECID
2119, 20sylibr 141 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 639  DECID wdc 753   wceq 1259   wcel 1409  c0 3252   csuc 4130  com 4341 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339 This theorem depends on definitions:  df-bi 114  df-dc 754  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342 This theorem is referenced by:  elni2  6470  indpi  6498
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