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Theorem nndceq0 4531
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 2146 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
21notbid 656 . . . 4  |-  ( x  =  (/)  ->  ( -.  x  =  (/)  <->  -.  (/)  =  (/) ) )
31, 2orbi12d 782 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( (/)  =  (/)  \/ 
-.  (/)  =  (/) ) ) )
4 eqeq1 2146 . . . 4  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
54notbid 656 . . . 4  |-  ( x  =  y  ->  ( -.  x  =  (/)  <->  -.  y  =  (/) ) )
64, 5orbi12d 782 . . 3  |-  ( x  =  y  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( y  =  (/)  \/ 
-.  y  =  (/) ) ) )
7 eqeq1 2146 . . . 4  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
87notbid 656 . . . 4  |-  ( x  =  suc  y  -> 
( -.  x  =  (/) 
<->  -.  suc  y  =  (/) ) )
97, 8orbi12d 782 . . 3  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) ) )
10 eqeq1 2146 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
1110notbid 656 . . . 4  |-  ( x  =  A  ->  ( -.  x  =  (/)  <->  -.  A  =  (/) ) )
1210, 11orbi12d 782 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) ) )
13 eqid 2139 . . . 4  |-  (/)  =  (/)
1413orci 720 . . 3  |-  ( (/)  =  (/)  \/  -.  (/)  =  (/) )
15 peano3 4510 . . . . . 6  |-  ( y  e.  om  ->  suc  y  =/=  (/) )
1615neneqd 2329 . . . . 5  |-  ( y  e.  om  ->  -.  suc  y  =  (/) )
1716olcd 723 . . . 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 4514 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
20 df-dc 820 . 2  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
2119, 20sylibr 133 1  |-  ( A  e.  om  -> DECID  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 697  DECID wdc 819    = wceq 1331    e. wcel 1480   (/)c0 3363   suc csuc 4287   omcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505
This theorem is referenced by:  omp1eomlem  6979  ctmlemr  6993  elni2  7134  indpi  7162  nnsf  13306  peano4nninf  13307
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