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

Theorem nnsuc 4633
Description: A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
nnsuc  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
Distinct variable group:    x, A

Proof of Theorem nnsuc
StepHypRef Expression
1 df-ne 2361 . 2  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 nn0suc 4621 . . . 4  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
32ord 725 . . 3  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  E. x  e.  om  A  =  suc  x ) )
43imp 124 . 2  |-  ( ( A  e.  om  /\  -.  A  =  (/) )  ->  E. x  e.  om  A  =  suc  x )
51, 4sylan2b 287 1  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    =/= wne 2360   E.wrex 2469   (/)c0 3437   suc csuc 4383   omcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608
This theorem is referenced by:  nnsucpred  4634
  Copyright terms: Public domain W3C validator