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Theorem nn0suc 3144
Description: A natural number is either 0 or a successor.
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 nnsuc 3138 . . . 4 |- ((A e. om /\ A =/= (/)) -> E.x e. om A = suc x)
2 df-ne 1579 . . . 4 |- (A =/= (/) <-> -. A = (/))
31, 2sylan2br 453 . . 3 |- ((A e. om /\ -. A = (/)) -> E.x e. om A = suc x)
43ex 373 . 2 |- (A e. om -> (-. A = (/) -> E.x e. om A = suc x))
54orrd 233 1 |- (A e. om -> (A = (/) \/ E.x e. om A = suc x))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   = wceq 953   e. wcel 955   =/= wne 1577  E.wrex 1638  (/)c0 2270  suc csuc 2940  omcom 3121
This theorem is referenced by:  nneneq 4492  php 4493
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122
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