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Theorem onunisuci 4543
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 4535 . 2  |-  Tr  A
31elexi 2831 . . 3  |-  A  e. 
_V
43unisuc 4505 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 199 1  |-  U. suc  A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   U.cuni 3864   Tr wtr 4150   Oncon0 4429   suc csuc 4431
This theorem is referenced by:  rankuni  7580  onsucconi  25262  onsucsuccmpi  25268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-v 2824  df-un 3191  df-in 3193  df-ss 3200  df-sn 3680  df-pr 3681  df-uni 3865  df-tr 4151  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435
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