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Theorem nn0suc 4355
Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
nn0suc  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Distinct variable group:    x, A

Proof of Theorem nn0suc
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . 3  |-  ( y  =  (/)  ->  ( y  =  (/)  <->  (/)  =  (/) ) )
2 eqeq1 2062 . . . 4  |-  ( y  =  (/)  ->  ( y  =  suc  x  <->  (/)  =  suc  x ) )
32rexbidv 2344 . . 3  |-  ( y  =  (/)  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  (/)  =  suc  x
) )
41, 3orbi12d 717 . 2  |-  ( y  =  (/)  ->  ( ( y  =  (/)  \/  E. x  e.  om  y  =  suc  x )  <->  ( (/)  =  (/)  \/ 
E. x  e.  om  (/)  =  suc  x ) ) )
5 eqeq1 2062 . . 3  |-  ( y  =  z  ->  (
y  =  (/)  <->  z  =  (/) ) )
6 eqeq1 2062 . . . 4  |-  ( y  =  z  ->  (
y  =  suc  x  <->  z  =  suc  x ) )
76rexbidv 2344 . . 3  |-  ( y  =  z  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  z  =  suc  x ) )
85, 7orbi12d 717 . 2  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( z  =  (/)  \/ 
E. x  e.  om  z  =  suc  x ) ) )
9 eqeq1 2062 . . 3  |-  ( y  =  suc  z  -> 
( y  =  (/)  <->  suc  z  =  (/) ) )
10 eqeq1 2062 . . . 4  |-  ( y  =  suc  z  -> 
( y  =  suc  x 
<->  suc  z  =  suc  x ) )
1110rexbidv 2344 . . 3  |-  ( y  =  suc  z  -> 
( E. x  e. 
om  y  =  suc  x 
<->  E. x  e.  om  suc  z  =  suc  x ) )
129, 11orbi12d 717 . 2  |-  ( y  =  suc  z  -> 
( ( y  =  (/)  \/  E. x  e. 
om  y  =  suc  x )  <->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) ) )
13 eqeq1 2062 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
14 eqeq1 2062 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
1514rexbidv 2344 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  A  =  suc  x ) )
1613, 15orbi12d 717 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( A  =  (/)  \/ 
E. x  e.  om  A  =  suc  x ) ) )
17 eqid 2056 . . 3  |-  (/)  =  (/)
1817orci 660 . 2  |-  ( (/)  =  (/)  \/  E. x  e.  om  (/)  =  suc  x
)
19 eqid 2056 . . . . 5  |-  suc  z  =  suc  z
20 suceq 4167 . . . . . . 7  |-  ( x  =  z  ->  suc  x  =  suc  z )
2120eqeq2d 2067 . . . . . 6  |-  ( x  =  z  ->  ( suc  z  =  suc  x 
<->  suc  z  =  suc  z ) )
2221rspcev 2673 . . . . 5  |-  ( ( z  e.  om  /\  suc  z  =  suc  z )  ->  E. x  e.  om  suc  z  =  suc  x )
2319, 22mpan2 409 . . . 4  |-  ( z  e.  om  ->  E. x  e.  om  suc  z  =  suc  x )
2423olcd 663 . . 3  |-  ( z  e.  om  ->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) )
2524a1d 22 . 2  |-  ( z  e.  om  ->  (
( z  =  (/)  \/ 
E. x  e.  om  z  =  suc  x )  ->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) ) )
264, 8, 12, 16, 18, 25finds 4351 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639    = wceq 1259    e. wcel 1409   E.wrex 2324   (/)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-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  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:  nnsuc  4366  nneneq  6351  phpm  6358  fin0  6373  fin0or  6374  diffisn  6381
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