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Theorem nnsuc 6496
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 6492 . . . 4  |-  ( A  e.  om  ->  -.  Lim  A )
21adantr 465 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  -.  Lim  A )
3 nnord 6487 . . . 4  |-  ( A  e.  om  ->  Ord  A )
4 orduninsuc 6457 . . . . . 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 4727 . . . . . . 7  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
76biimpri 206 . . . . . 6  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
873expia 1189 . . . . 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 2503 . . . . . . . 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 6495 . . . . . . 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 2828 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2609   E.wrex 2719   (/)c0 3640   U.cuni 4094   Ord word 4721   Oncon0 4722   Lim wlim 4723   suc csuc 4724   omcom 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-tr 4389  df-eprel 4635  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-om 6480
This theorem is referenced by:  peano5  6502  nn0suc  6503  inf3lemd  7836  infpssrlem4  8478  fin1a2lem6  8577  bnj158  31723  bnj1098  31780  bnj594  31908
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