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Theorem ordunisuc 4779
Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ordunisuc  |-  ( Ord 
A  ->  U. suc  A  =  A )

Proof of Theorem ordunisuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordeleqon 4736 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4614 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 3994 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2426 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4559 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4563 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 16 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 2927 . . . . . 6  |-  x  e. 
_V
109unisuc 4625 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 189 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 2985 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 4755 . . . . . 6  |-  suc  On  =  On
1413unieqi 3993 . . . . 5  |-  U. suc  On  =  U. On
15 unon 4778 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2432 . . . 4  |-  U. suc  On  =  On
17 suceq 4614 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 3994 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 20 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2470 . . 3  |-  ( A  =  On  ->  U. suc  A  =  A )
2112, 20jaoi 369 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  U. suc  A  =  A )
221, 21sylbi 188 1  |-  ( Ord 
A  ->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1721   U.cuni 3983   Tr wtr 4270   Ord word 4548   Oncon0 4549   suc csuc 4551
This theorem is referenced by:  orduniss2  4780  onsucuni2  4781  nlimsucg  4789  tz7.44-2  6632  ttukeylem7  8359  tsksuc  8601  dfrdg2  25374  ontgsucval  26094  onsuctopon  26096  limsucncmpi  26107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555
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