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Theorem nn0suc 4696
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 2461 . . . 4  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 nnsuc 4689 . . . 4  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
31, 2sylan2br 462 . . 3  |-  ( ( A  e.  om  /\  -.  A  =  (/) )  ->  E. x  e.  om  A  =  suc  x )
43ex 423 . 2  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  E. x  e.  om  A  =  suc  x ) )
54orrd 367 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468   suc csuc 4410   omcom 4672
This theorem is referenced by:  nnawordex  6651  nneneq  7060  php  7061  cantnfvalf  7382  cantnflt  7389  hsmexlem9  8067  winainflem  8331  trpredlem1  24301  trpred0  24310  trpredrec  24312  bnj517  29233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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