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Theorem onunisuci 4117
Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
on.1 A On
Assertion
Ref Expression
onunisuci suc A = A

Proof of Theorem onunisuci
StepHypRef Expression
1 on.1 . . 3 A On
21ontrci 4112 . 2 Tr A
31elexi 2543 . . 3 A V
43unisuc 4097 . 2 (Tr A suc A = A)
52, 4mpbi 133 1 suc A = A
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1375   cuni 3553  Tr wtr 3827  Oncon0 4047  suc csuc 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055
This theorem is referenced by: (None)
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