ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nndceq0 Unicode version

Theorem nndceq0 4431
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 2094 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
21notbid 627 . . . 4  |-  ( x  =  (/)  ->  ( -.  x  =  (/)  <->  -.  (/)  =  (/) ) )
31, 2orbi12d 742 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( (/)  =  (/)  \/ 
-.  (/)  =  (/) ) ) )
4 eqeq1 2094 . . . 4  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
54notbid 627 . . . 4  |-  ( x  =  y  ->  ( -.  x  =  (/)  <->  -.  y  =  (/) ) )
64, 5orbi12d 742 . . 3  |-  ( x  =  y  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( y  =  (/)  \/ 
-.  y  =  (/) ) ) )
7 eqeq1 2094 . . . 4  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
87notbid 627 . . . 4  |-  ( x  =  suc  y  -> 
( -.  x  =  (/) 
<->  -.  suc  y  =  (/) ) )
97, 8orbi12d 742 . . 3  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) ) )
10 eqeq1 2094 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
1110notbid 627 . . . 4  |-  ( x  =  A  ->  ( -.  x  =  (/)  <->  -.  A  =  (/) ) )
1210, 11orbi12d 742 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) ) )
13 eqid 2088 . . . 4  |-  (/)  =  (/)
1413orci 685 . . 3  |-  ( (/)  =  (/)  \/  -.  (/)  =  (/) )
15 peano3 4411 . . . . . 6  |-  ( y  e.  om  ->  suc  y  =/=  (/) )
1615neneqd 2276 . . . . 5  |-  ( y  e.  om  ->  -.  suc  y  =  (/) )
1716olcd 688 . . . 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 4415 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
20 df-dc 781 . 2  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
2119, 20sylibr 132 1  |-  ( A  e.  om  -> DECID  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   (/)c0 3286   suc csuc 4192   omcom 4405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-suc 4198  df-iom 4406
This theorem is referenced by:  elni2  6871  indpi  6899  nnsf  11850  peano4nninf  11851
  Copyright terms: Public domain W3C validator