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Theorem ordunisuc 6662
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 6619 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4949 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 4261 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2489 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4894 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4898 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 16 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 3121 . . . . . 6  |-  x  e. 
_V
109unisuc 4960 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 196 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 3182 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 6638 . . . . . 6  |-  suc  On  =  On
1413unieqi 4260 . . . . 5  |-  U. suc  On  =  U. On
15 unon 6661 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2496 . . . 4  |-  U. suc  On  =  On
17 suceq 4949 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 4261 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 22 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2534 . . 3  |-  ( A  =  On  ->  U. suc  A  =  A )
2112, 20jaoi 379 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  U. suc  A  =  A )
221, 21sylbi 195 1  |-  ( Ord 
A  ->  U. suc  A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767   U.cuni 4251   Tr wtr 4546   Ord word 4883   Oncon0 4884   suc csuc 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890
This theorem is referenced by:  orduniss2  6663  onsucuni2  6664  nlimsucg  6672  tz7.44-2  7085  ttukeylem7  8907  tsksuc  9152  dfrdg2  29155  ontgsucval  29824  onsuctopon  29826  limsucncmpi  29837
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