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Theorem nnsuc 4673
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 4669 . . . 4  |-  ( A  e.  om  ->  -.  Lim  A )
21adantr 451 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  -.  Lim  A )
3 nnord 4664 . . . 4  |-  ( A  e.  om  ->  Ord  A )
4 orduninsuc 4634 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
54adantr 451 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
6 df-lim 4397 . . . . . . 7  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
76biimpri 197 . . . . . 6  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
873expia 1153 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  ->  Lim  A ) )
95, 8sylbird 226 . . . 4  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
103, 9sylan 457 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  Lim  A ) )
112, 10mt3d 117 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  On  A  =  suc  x )
12 eleq1 2343 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( A  e.  om  <->  suc  x  e.  om )
)
1312biimpcd 215 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  suc  x  e.  om )
)
14 peano2b 4672 . . . . . . 7  |-  ( x  e.  om  <->  suc  x  e. 
om )
1513, 14syl6ibr 218 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  x  e.  om )
)
1615ancrd 537 . . . . 5  |-  ( A  e.  om  ->  ( A  =  suc  x  -> 
( x  e.  om  /\  A  =  suc  x
) ) )
1716adantld 453 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  On  /\  A  =  suc  x
)  ->  ( x  e.  om  /\  A  =  suc  x ) ) )
1817reximdv2 2652 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
1918adantr 451 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
2011, 19mpd 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   U.cuni 3827   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656
This theorem is referenced by:  peano5  4679  nn0suc  4680  inf3lemd  7328  infpssrlem4  7932  fin1a2lem6  8031  bnj158  28757  bnj1098  28815  bnj594  28944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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