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Theorem ordunisuc 6679
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 6635 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 5513 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 4235 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 23 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2445 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 5458 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 5462 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 17 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 3088 . . . . . 6  |-  x  e. 
_V
109unisuc 5524 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 200 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 3151 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 6655 . . . . . 6  |-  suc  On  =  On
1413unieqi 4234 . . . . 5  |-  U. suc  On  =  U. On
15 unon 6678 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2452 . . . 4  |-  U. suc  On  =  On
17 suceq 5513 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 4235 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 23 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2490 . . 3  |-  ( A  =  On  ->  U. suc  A  =  A )
2112, 20jaoi 381 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  U. suc  A  =  A )
221, 21sylbi 199 1  |-  ( Ord 
A  ->  U. suc  A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    = wceq 1438    e. wcel 1873   U.cuni 4225   Tr wtr 4524   Ord word 5447   Oncon0 5448   suc csuc 5450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pr 4666  ax-un 6603
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3087  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-br 4430  df-opab 4489  df-tr 4525  df-eprel 4770  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-ord 5451  df-on 5452  df-suc 5454
This theorem is referenced by:  orduniss2  6680  onsucuni2  6681  nlimsucg  6689  tz7.44-2  7142  ttukeylem7  8958  tsksuc  9200  dfrdg2  30455  ontgsucval  31103  onsuctopon  31105  limsucncmpi  31116  finxpsuclem  31759
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