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Theorem nn0suc 4586
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 2177 . . 3  |-  ( y  =  (/)  ->  ( y  =  (/)  <->  (/)  =  (/) ) )
2 eqeq1 2177 . . . 4  |-  ( y  =  (/)  ->  ( y  =  suc  x  <->  (/)  =  suc  x ) )
32rexbidv 2471 . . 3  |-  ( y  =  (/)  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  (/)  =  suc  x
) )
41, 3orbi12d 788 . 2  |-  ( y  =  (/)  ->  ( ( y  =  (/)  \/  E. x  e.  om  y  =  suc  x )  <->  ( (/)  =  (/)  \/ 
E. x  e.  om  (/)  =  suc  x ) ) )
5 eqeq1 2177 . . 3  |-  ( y  =  z  ->  (
y  =  (/)  <->  z  =  (/) ) )
6 eqeq1 2177 . . . 4  |-  ( y  =  z  ->  (
y  =  suc  x  <->  z  =  suc  x ) )
76rexbidv 2471 . . 3  |-  ( y  =  z  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  z  =  suc  x ) )
85, 7orbi12d 788 . 2  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( z  =  (/)  \/ 
E. x  e.  om  z  =  suc  x ) ) )
9 eqeq1 2177 . . 3  |-  ( y  =  suc  z  -> 
( y  =  (/)  <->  suc  z  =  (/) ) )
10 eqeq1 2177 . . . 4  |-  ( y  =  suc  z  -> 
( y  =  suc  x 
<->  suc  z  =  suc  x ) )
1110rexbidv 2471 . . 3  |-  ( y  =  suc  z  -> 
( E. x  e. 
om  y  =  suc  x 
<->  E. x  e.  om  suc  z  =  suc  x ) )
129, 11orbi12d 788 . 2  |-  ( y  =  suc  z  -> 
( ( y  =  (/)  \/  E. x  e. 
om  y  =  suc  x )  <->  ( suc  z  =  (/)  \/  E. x  e.  om  suc  z  =  suc  x ) ) )
13 eqeq1 2177 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
14 eqeq1 2177 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
1514rexbidv 2471 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  =  suc  x  <->  E. x  e.  om  A  =  suc  x ) )
1613, 15orbi12d 788 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  om  y  =  suc  x )  <-> 
( A  =  (/)  \/ 
E. x  e.  om  A  =  suc  x ) ) )
17 eqid 2170 . . 3  |-  (/)  =  (/)
1817orci 726 . 2  |-  ( (/)  =  (/)  \/  E. x  e.  om  (/)  =  suc  x
)
19 eqid 2170 . . . . 5  |-  suc  z  =  suc  z
20 suceq 4385 . . . . . . 7  |-  ( x  =  z  ->  suc  x  =  suc  z )
2120eqeq2d 2182 . . . . . 6  |-  ( x  =  z  ->  ( suc  z  =  suc  x 
<->  suc  z  =  suc  z ) )
2221rspcev 2834 . . . . 5  |-  ( ( z  e.  om  /\  suc  z  =  suc  z )  ->  E. x  e.  om  suc  z  =  suc  x )
2319, 22mpan2 423 . . . 4  |-  ( z  e.  om  ->  E. x  e.  om  suc  z  =  suc  x )
2423olcd 729 . . 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 4582 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 703    = wceq 1348    e. wcel 2141   E.wrex 2449   (/)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-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  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:  nnsuc  4598  nnpredcl  4605  frecabcl  6375  nnsucuniel  6471  nneneq  6831  phpm  6839  dif1enen  6854  fin0  6859  fin0or  6860  diffisn  6867
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