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Theorem onunisuci 4195
Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onunisuci  |-  U. suc  A  =  A

Proof of Theorem onunisuci
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21ontrci 4190 . 2  |-  Tr  A
31elexi 2612 . . 3  |-  A  e. 
_V
43unisuc 4176 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 143 1  |-  U. suc  A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   U.cuni 3609   Tr wtr 3883   Oncon0 4126   suc csuc 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134
This theorem is referenced by: (None)
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