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Theorem nn0suc 4432
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 2095 . . 3  |-  ( y  =  (/)  ->  ( y  =  (/)  <->  (/)  =  (/) ) )
2 eqeq1 2095 . . . 4  |-  ( y  =  (/)  ->  ( y  =  suc  x  <->  (/)  =  suc  x ) )
32rexbidv 2382 . . 3  |-  ( y  =  (/)  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  (/)  =  suc  x
) )
41, 3orbi12d 743 . 2  |-  ( y  =  (/)  ->  ( ( y  =  (/)  \/  E. x  e.  om  y  =  suc  x )  <->  ( (/)  =  (/)  \/ 
E. x  e.  om  (/)  =  suc  x ) ) )
5 eqeq1 2095 . . 3  |-  ( y  =  z  ->  (
y  =  (/)  <->  z  =  (/) ) )
6 eqeq1 2095 . . . 4  |-  ( y  =  z  ->  (
y  =  suc  x  <->  z  =  suc  x ) )
76rexbidv 2382 . . 3  |-  ( y  =  z  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  z  =  suc  x ) )
85, 7orbi12d 743 . 2  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( z  =  (/)  \/ 
E. x  e.  om  z  =  suc  x ) ) )
9 eqeq1 2095 . . 3  |-  ( y  =  suc  z  -> 
( y  =  (/)  <->  suc  z  =  (/) ) )
10 eqeq1 2095 . . . 4  |-  ( y  =  suc  z  -> 
( y  =  suc  x 
<->  suc  z  =  suc  x ) )
1110rexbidv 2382 . . 3  |-  ( y  =  suc  z  -> 
( E. x  e. 
om  y  =  suc  x 
<->  E. x  e.  om  suc  z  =  suc  x ) )
129, 11orbi12d 743 . 2  |-  ( y  =  suc  z  -> 
( ( y  =  (/)  \/  E. x  e. 
om  y  =  suc  x )  <->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) ) )
13 eqeq1 2095 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
14 eqeq1 2095 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
1514rexbidv 2382 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  A  =  suc  x ) )
1613, 15orbi12d 743 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( A  =  (/)  \/ 
E. x  e.  om  A  =  suc  x ) ) )
17 eqid 2089 . . 3  |-  (/)  =  (/)
1817orci 686 . 2  |-  ( (/)  =  (/)  \/  E. x  e.  om  (/)  =  suc  x
)
19 eqid 2089 . . . . 5  |-  suc  z  =  suc  z
20 suceq 4238 . . . . . . 7  |-  ( x  =  z  ->  suc  x  =  suc  z )
2120eqeq2d 2100 . . . . . 6  |-  ( x  =  z  ->  ( suc  z  =  suc  x 
<->  suc  z  =  suc  z ) )
2221rspcev 2723 . . . . 5  |-  ( ( z  e.  om  /\  suc  z  =  suc  z )  ->  E. x  e.  om  suc  z  =  suc  x )
2319, 22mpan2 417 . . . 4  |-  ( z  e.  om  ->  E. x  e.  om  suc  z  =  suc  x )
2423olcd 689 . . 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 4428 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 665    = wceq 1290    e. wcel 1439   E.wrex 2361   (/)c0 3287   suc csuc 4201   omcom 4418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-suc 4207  df-iom 4419
This theorem is referenced by:  nnsuc  4443  nnpredcl  4449  frecabcl  6178  nnsucuniel  6270  nneneq  6627  phpm  6635  dif1enen  6650  fin0  6655  fin0or  6656  diffisn  6663
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