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Theorem onunisuci 4695
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 4687 . 2  |-  Tr  A
31elexi 2965 . . 3  |-  A  e. 
_V
43unisuc 4657 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 200 1  |-  U. suc  A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   U.cuni 4015   Tr wtr 4302   Oncon0 4581   suc csuc 4583
This theorem is referenced by:  rankuni  7789  onsucconi  26187  onsucsuccmpi  26193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-sn 3820  df-pr 3821  df-uni 4016  df-tr 4303  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587
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