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Theorem nnsuc 6698
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 6694 . . . 4  |-  ( A  e.  om  ->  -.  Lim  A )
21adantr 465 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  -.  Lim  A )
3 nnord 6689 . . . 4  |-  ( A  e.  om  ->  Ord  A )
4 orduninsuc 6659 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
54adantr 465 . . . . 5  |-  ( ( Ord  A  /\  A  =/=  (/) )  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
6 df-lim 4869 . . . . . . 7  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
76biimpri 206 . . . . . 6  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
873expia 1197 . . . . 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 471 . . 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 2513 . . . . . . . 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 6697 . . . . . . 7  |-  ( x  e.  om  <->  suc  x  e. 
om )
1513, 14syl6ibr 227 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  suc  x  ->  x  e.  om )
)
1615ancrd 554 . . . . 5  |-  ( A  e.  om  ->  ( A  =  suc  x  -> 
( x  e.  om  /\  A  =  suc  x
) ) )
1716adantld 467 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  On  /\  A  =  suc  x
)  ->  ( x  e.  om  /\  A  =  suc  x ) ) )
1817reximdv2 2912 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  On  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
1918adantr 465 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792   (/)c0 3767   U.cuni 4230   Ord word 4863   Oncon0 4864   Lim wlim 4865   suc csuc 4866   omcom 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-om 6682
This theorem is referenced by:  peano5  6704  nn0suc  6705  inf3lemd  8042  infpssrlem4  8684  fin1a2lem6  8783  bnj158  33492  bnj1098  33549  bnj594  33677
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