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Theorem onunisuci 4324
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 4319 . 2  |-  Tr  A
31elexi 2672 . . 3  |-  A  e. 
_V
43unisuc 4305 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 144 1  |-  U. suc  A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   U.cuni 3706   Tr wtr 3996   Oncon0 4255   suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by: (None)
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