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Theorem nsuceq0 3043
Description: No successor is empty.
Assertion
Ref Expression
nsuceq0 |- suc A =/= (/)
Proof of Theorem nsuceq0
StepHypRef Expression
1 noel 2274 . . . 4 |- -. A e. (/)
2 eleq2 1527 . . . . 5 |- (suc A = (/) -> (A e. suc A <-> A e. (/)))
3 sucidg 3042 . . . . 5 |- (A e. V -> A e. suc A)
42, 3syl5cbi 209 . . . 4 |- (A e. V -> (suc A = (/) -> A e. (/)))
51, 4mtoi 107 . . 3 |- (A e. V -> -. suc A = (/))
6 sucprc 3034 . . . . . . 7 |- (-. A e. V -> suc A = A)
76eqeq1d 1475 . . . . . 6 |- (-. A e. V -> (suc A = (/) <-> A = (/)))
8 0ex 2701 . . . . . . 7 |- (/) e. V
9 eleq1 1526 . . . . . . 7 |- (A = (/) -> (A e. V <-> (/) e. V))
108, 9mpbiri 194 . . . . . 6 |- (A = (/) -> A e. V)
117, 10syl6bi 214 . . . . 5 |- (-. A e. V -> (suc A = (/) -> A e. V))
1211con3d 95 . . . 4 |- (-. A e. V -> (-. A e. V -> -. suc A = (/)))
1312pm2.43i 64 . . 3 |- (-. A e. V -> -. suc A = (/))
145, 13pm2.61i 126 . 2 |- -. suc A = (/)
15 df-ne 1579 . 2 |- (suc A =/= (/) <-> -. suc A = (/))
1614, 15mpbir 190 1 |- suc A =/= (/)
Colors of variables: wff set class
Syntax hints:  -. wn 2   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802  (/)c0 2270  suc csuc 2940
This theorem is referenced by:  0elsuc 3082  peano3 3141  tz7.44-2 3914  oelim2 4206  limenpsi 4485  cfsuc 4887
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-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-nul 2700
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403  df-suc 2944
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