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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  |-  ( A  e.  om  -> DECID  A  =  (/) )

Proof of Theorem nndceq0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
21notbid 602 . . . 4  |-  ( x  =  (/)  ->  ( -.  x  =  (/)  <->  -.  (/)  =  (/) ) )
31, 2orbi12d 717 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( (/)  =  (/)  \/ 
-.  (/)  =  (/) ) ) )
4 eqeq1 2062 . . . 4  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
54notbid 602 . . . 4  |-  ( x  =  y  ->  ( -.  x  =  (/)  <->  -.  y  =  (/) ) )
64, 5orbi12d 717 . . 3  |-  ( x  =  y  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( y  =  (/)  \/ 
-.  y  =  (/) ) ) )
7 eqeq1 2062 . . . 4  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
87notbid 602 . . . 4  |-  ( x  =  suc  y  -> 
( -.  x  =  (/) 
<->  -.  suc  y  =  (/) ) )
97, 8orbi12d 717 . . 3  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) ) )
10 eqeq1 2062 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
1110notbid 602 . . . 4  |-  ( x  =  A  ->  ( -.  x  =  (/)  <->  -.  A  =  (/) ) )
1210, 11orbi12d 717 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) ) )
13 eqid 2056 . . . 4  |-  (/)  =  (/)
1413orci 660 . . 3  |-  ( (/)  =  (/)  \/  -.  (/)  =  (/) )
15 peano3 4347 . . . . . 6  |-  ( y  e.  om  ->  suc  y  =/=  (/) )
1615neneqd 2241 . . . . 5  |-  ( y  e.  om  ->  -.  suc  y  =  (/) )
1716olcd 663 . . . 4  |-  ( y  e.  om  ->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) )
1817a1d 22 . . 3  |-  ( y  e.  om  ->  (
( y  =  (/)  \/ 
-.  y  =  (/) )  ->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) ) )
193, 6, 9, 12, 14, 18finds 4351 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
20 df-dc 754 . 2  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
2119, 20sylibr 141 1  |-  ( A  e.  om  -> DECID  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 639  DECID wdc 753    = wceq 1259    e. wcel 1409   (/)c0 3252   suc csuc 4130   omcom 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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