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Theorem onunisuci 4943
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 4935 . 2  |-  Tr  A
31elexi 3088 . . 3  |-  A  e. 
_V
43unisuc 4906 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 208 1  |-  U. suc  A  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   U.cuni 4202   Tr wtr 4496   Oncon0 4830   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-pr 3991  df-uni 4203  df-tr 4497  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836
This theorem is referenced by:  rankuni  8184  onsucconi  28447  onsucsuccmpi  28453
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