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Theorem onunisuci 4932
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 4924 . 2  |-  Tr  A
31elexi 3066 . . 3  |-  A  e. 
_V
43unisuc 4895 . 2  |-  ( Tr  A  <->  U. suc  A  =  A )
52, 4mpbi 208 1  |-  U. suc  A  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403    e. wcel 1840   U.cuni 4188   Tr wtr 4486   Oncon0 4819   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-v 3058  df-un 3416  df-in 3418  df-ss 3425  df-sn 3970  df-pr 3972  df-uni 4189  df-tr 4487  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825
This theorem is referenced by:  rankuni  8231  onsucconi  30652  onsucsuccmpi  30658
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