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

Theorem nndceq0 4600
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 2177 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
21notbid 662 . . . 4  |-  ( x  =  (/)  ->  ( -.  x  =  (/)  <->  -.  (/)  =  (/) ) )
31, 2orbi12d 788 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( (/)  =  (/)  \/ 
-.  (/)  =  (/) ) ) )
4 eqeq1 2177 . . . 4  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
54notbid 662 . . . 4  |-  ( x  =  y  ->  ( -.  x  =  (/)  <->  -.  y  =  (/) ) )
64, 5orbi12d 788 . . 3  |-  ( x  =  y  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( y  =  (/)  \/ 
-.  y  =  (/) ) ) )
7 eqeq1 2177 . . . 4  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
87notbid 662 . . . 4  |-  ( x  =  suc  y  -> 
( -.  x  =  (/) 
<->  -.  suc  y  =  (/) ) )
97, 8orbi12d 788 . . 3  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  -.  x  =  (/) )  <->  ( suc  y  =  (/)  \/  -.  suc  y  =  (/) ) ) )
10 eqeq1 2177 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
1110notbid 662 . . . 4  |-  ( x  =  A  ->  ( -.  x  =  (/)  <->  -.  A  =  (/) ) )
1210, 11orbi12d 788 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/ 
-.  x  =  (/) ) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) ) )
13 eqid 2170 . . . 4  |-  (/)  =  (/)
1413orci 726 . . 3  |-  ( (/)  =  (/)  \/  -.  (/)  =  (/) )
15 peano3 4578 . . . . . 6  |-  ( y  e.  om  ->  suc  y  =/=  (/) )
1615neneqd 2361 . . . . 5  |-  ( y  e.  om  ->  -.  suc  y  =  (/) )
1716olcd 729 . . . 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 4582 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
20 df-dc 830 . 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 703  DECID wdc 829    = wceq 1348    e. wcel 2141   (/)c0 3414   suc csuc 4348   omcom 4572
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573
This theorem is referenced by:  omp1eomlem  7068  ctmlemr  7082  nnnninfeq2  7102  nninfisol  7106  elni2  7265  indpi  7293  nnsf  14000  peano4nninf  14001
  Copyright terms: Public domain W3C validator