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Theorem nnsuc 3138
Description: A non-zero natural number is a successor.
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 3134 . . . 4 |- (A e. om -> -. Lim A)
21adantr 389 . . 3 |- ((A e. om /\ A =/= (/)) -> -. Lim A)
3 orduninsuc 3104 . . . . . 6 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
43adantr 389 . . . . 5 |- ((Ord A /\ A =/= (/)) -> (A = U.A <-> -. E.x e. On A = suc x))
5 df-lim 2943 . . . . . . 7 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
65biimpr 152 . . . . . 6 |- ((Ord A /\ A =/= (/) /\ A = U.A) -> Lim A)
763expia 833 . . . . 5 |- ((Ord A /\ A =/= (/)) -> (A = U.A -> Lim A))
84, 7sylbird 205 . . . 4 |- ((Ord A /\ A =/= (/)) -> (-. E.x e. On A = suc x -> Lim A))
9 nnord 3130 . . . 4 |- (A e. om -> Ord A)
108, 9sylan 448 . . 3 |- ((A e. om /\ A =/= (/)) -> (-. E.x e. On A = suc x -> Lim A))
112, 10mt3d 114 . 2 |- ((A e. om /\ A =/= (/)) -> E.x e. On A = suc x)
12 eleq1 1526 . . . . . . . . 9 |- (A = suc x -> (A e. om <-> suc x e. om))
1312biimpcd 155 . . . . . . . 8 |- (A e. om -> (A = suc x -> suc x e. om))
14 peano2b 3137 . . . . . . . 8 |- (x e. om <-> suc x e. om)
1513, 14syl6ibr 213 . . . . . . 7 |- (A e. om -> (A = suc x -> x e. om))
1615ancrd 299 . . . . . 6 |- (A e. om -> (A = suc x -> (x e. om /\ A = suc x)))
1716adantld 390 . . . . 5 |- (A e. om -> ((x e. On /\ A = suc x) -> (x e. om /\ A = suc x)))
181719.22dv 1285 . . . 4 |- (A e. om -> (E.x(x e. On /\ A = suc x) -> E.x(x e. om /\ A = suc x)))
19 df-rex 1642 . . . 4 |- (E.x e. On A = suc x <-> E.x(x e. On /\ A = suc x))
20 df-rex 1642 . . . 4 |- (E.x e. om A = suc x <-> E.x(x e. om /\ A = suc x))
2118, 19, 203imtr4g 551 . . 3 |- (A e. om -> (E.x e. On A = suc x -> E.x e. om A = suc x))
2221adantr 389 . 2 |- ((A e. om /\ A =/= (/)) -> (E.x e. On A = suc x -> E.x e. om A = suc x))
2311, 22mpd 26 1 |- ((A e. om /\ A =/= (/)) -> E.x e. om A = suc x)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  E.wrex 1638  (/)c0 2270  U.cuni 2493  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940  omcom 3121
This theorem is referenced by:  peano5 3143  nn0suc 3144  inf3lemd 4584
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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