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Theorem nn0suc 6500
Description: A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
nn0suc  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Distinct variable group:    x, A

Proof of Theorem nn0suc
StepHypRef Expression
1 df-ne 2608 . . . 4  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 nnsuc 6493 . . . 4  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
31, 2sylan2br 476 . . 3  |-  ( ( A  e.  om  /\  -.  A  =  (/) )  ->  E. x  e.  om  A  =  suc  x )
43ex 434 . 2  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  E. x  e.  om  A  =  suc  x ) )
54orrd 378 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   (/)c0 3637   suc csuc 4721   omcom 6476
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 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-om 6477
This theorem is referenced by:  nnawordex  7076  nneneq  7494  php  7495  cantnfvalf  7873  cantnflt  7880  cantnfltOLD  7910  hsmexlem9  8594  winainflem  8860  trpredlem1  27691  trpred0  27700  trpredrec  27702  bnj517  31878
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