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Theorem nnsuc 4854
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 4850 . . . 4  |-  ( A  e.  om  ->  -.  Lim  A )
21adantr 452 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  -.  Lim  A )
3 nnord 4845 . . . 4  |-  ( A  e.  om  ->  Ord  A )
4 orduninsuc 4815 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
54adantr 452 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
6 df-lim 4578 . . . . . . 7  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
76biimpri 198 . . . . . 6  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
873expia 1155 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  ->  Lim  A ) )
95, 8sylbird 227 . . . 4  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
103, 9sylan 458 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
112, 10mt3d 119 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  On  A  =  suc  x )
12 eleq1 2495 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( A  e.  om  <->  suc  x  e.  om )
)
1312biimpcd 216 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  suc  x  e.  om )
)
14 peano2b 4853 . . . . . . 7  |-  ( x  e.  om  <->  suc  x  e. 
om )
1513, 14syl6ibr 219 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  x  e.  om )
)
1615ancrd 538 . . . . 5  |-  ( A  e.  om  ->  ( A  =  suc  x  -> 
( x  e.  om  /\  A  =  suc  x
) ) )
1716adantld 454 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  On  /\  A  =  suc  x
)  ->  ( x  e.  om  /\  A  =  suc  x ) ) )
1817reximdv2 2807 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
1918adantr 452 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   (/)c0 3620   U.cuni 4007   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   omcom 4837
This theorem is referenced by:  peano5  4860  nn0suc  4861  inf3lemd  7574  infpssrlem4  8178  fin1a2lem6  8277  bnj158  29033  bnj1098  29091  bnj594  29220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838
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