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Theorem ordunisuc 4726
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 4683 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4560 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 3940 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2380 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4505 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4509 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 15 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 2876 . . . . . 6  |-  x  e. 
_V
109unisuc 4571 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 188 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 2934 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 4702 . . . . . 6  |-  suc  On  =  On
1413unieqi 3939 . . . . 5  |-  U. suc  On  =  U. On
15 unon 4725 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2386 . . . 4  |-  U. suc  On  =  On
17 suceq 4560 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 3940 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 19 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2424 . . 3  |-  ( A  =  On  ->  U. suc  A  =  A )
2112, 20jaoi 368 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  U. suc  A  =  A )
221, 21sylbi 187 1  |-  ( Ord 
A  ->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1647    e. wcel 1715   U.cuni 3929   Tr wtr 4215   Ord word 4494   Oncon0 4495   suc csuc 4497
This theorem is referenced by:  orduniss2  4727  onsucuni2  4728  nlimsucg  4736  tz7.44-2  6562  ttukeylem7  8289  tsksuc  8531  dfrdg2  24978  ontgsucval  25698  onsuctopon  25700  limsucncmpi  25711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-suc 4501
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