MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnsuc Structured version   Unicode version

Theorem nnsuc 6492
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 nnlim 6488 . . . 4  |-  ( A  e.  om  ->  -.  Lim  A )
21adantr 462 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  -.  Lim  A )
3 nnord 6483 . . . 4  |-  ( A  e.  om  ->  Ord  A )
4 orduninsuc 6453 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
54adantr 462 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
6 df-lim 4720 . . . . . . 7  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
76biimpri 206 . . . . . 6  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
873expia 1184 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  ->  Lim  A ) )
95, 8sylbird 235 . . . 4  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
103, 9sylan 468 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
112, 10mt3d 125 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  On  A  =  suc  x )
12 eleq1 2501 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( A  e.  om  <->  suc  x  e.  om )
)
1312biimpcd 224 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  suc  x  e.  om )
)
14 peano2b 6491 . . . . . . 7  |-  ( x  e.  om  <->  suc  x  e. 
om )
1513, 14syl6ibr 227 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  x  e.  om )
)
1615ancrd 551 . . . . 5  |-  ( A  e.  om  ->  ( A  =  suc  x  -> 
( x  e.  om  /\  A  =  suc  x
) ) )
1716adantld 464 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  On  /\  A  =  suc  x
)  ->  ( x  e.  om  /\  A  =  suc  x ) ) )
1817reximdv2 2823 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
1918adantr 462 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
2011, 19mpd 15 1  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   (/)c0 3634   U.cuni 4088   Ord word 4714   Oncon0 4715   Lim wlim 4716   suc csuc 4717   omcom 6475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-om 6476
This theorem is referenced by:  peano5  6498  nn0suc  6499  inf3lemd  7829  infpssrlem4  8471  fin1a2lem6  8570  bnj158  31554  bnj1098  31611  bnj594  31739
  Copyright terms: Public domain W3C validator