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Theorem nn0suc 4513
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 2144 . . 3  |-  ( y  =  (/)  ->  ( y  =  (/)  <->  (/)  =  (/) ) )
2 eqeq1 2144 . . . 4  |-  ( y  =  (/)  ->  ( y  =  suc  x  <->  (/)  =  suc  x ) )
32rexbidv 2436 . . 3  |-  ( y  =  (/)  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  (/)  =  suc  x
) )
41, 3orbi12d 782 . 2  |-  ( y  =  (/)  ->  ( ( y  =  (/)  \/  E. x  e.  om  y  =  suc  x )  <->  ( (/)  =  (/)  \/ 
E. x  e.  om  (/)  =  suc  x ) ) )
5 eqeq1 2144 . . 3  |-  ( y  =  z  ->  (
y  =  (/)  <->  z  =  (/) ) )
6 eqeq1 2144 . . . 4  |-  ( y  =  z  ->  (
y  =  suc  x  <->  z  =  suc  x ) )
76rexbidv 2436 . . 3  |-  ( y  =  z  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  z  =  suc  x ) )
85, 7orbi12d 782 . 2  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( z  =  (/)  \/ 
E. x  e.  om  z  =  suc  x ) ) )
9 eqeq1 2144 . . 3  |-  ( y  =  suc  z  -> 
( y  =  (/)  <->  suc  z  =  (/) ) )
10 eqeq1 2144 . . . 4  |-  ( y  =  suc  z  -> 
( y  =  suc  x 
<->  suc  z  =  suc  x ) )
1110rexbidv 2436 . . 3  |-  ( y  =  suc  z  -> 
( E. x  e. 
om  y  =  suc  x 
<->  E. x  e.  om  suc  z  =  suc  x ) )
129, 11orbi12d 782 . 2  |-  ( y  =  suc  z  -> 
( ( y  =  (/)  \/  E. x  e. 
om  y  =  suc  x )  <->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) ) )
13 eqeq1 2144 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
14 eqeq1 2144 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
1514rexbidv 2436 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  A  =  suc  x ) )
1613, 15orbi12d 782 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( A  =  (/)  \/ 
E. x  e.  om  A  =  suc  x ) ) )
17 eqid 2137 . . 3  |-  (/)  =  (/)
1817orci 720 . 2  |-  ( (/)  =  (/)  \/  E. x  e.  om  (/)  =  suc  x
)
19 eqid 2137 . . . . 5  |-  suc  z  =  suc  z
20 suceq 4319 . . . . . . 7  |-  ( x  =  z  ->  suc  x  =  suc  z )
2120eqeq2d 2149 . . . . . 6  |-  ( x  =  z  ->  ( suc  z  =  suc  x 
<->  suc  z  =  suc  z ) )
2221rspcev 2784 . . . . 5  |-  ( ( z  e.  om  /\  suc  z  =  suc  z )  ->  E. x  e.  om  suc  z  =  suc  x )
2319, 22mpan2 421 . . . 4  |-  ( z  e.  om  ->  E. x  e.  om  suc  z  =  suc  x )
2423olcd 723 . . 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 4509 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 697    = wceq 1331    e. wcel 1480   E.wrex 2415   (/)c0 3358   suc csuc 4282   omcom 4499
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 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500
This theorem is referenced by:  nnsuc  4524  nnpredcl  4531  frecabcl  6289  nnsucuniel  6384  nneneq  6744  phpm  6752  dif1enen  6767  fin0  6772  fin0or  6773  diffisn  6780
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