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Theorem ordunisuc 6666
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 6623 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4952 . . . . . 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 2479 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4897 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4901 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 16 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 3112 . . . . . 6  |-  x  e. 
_V
109unisuc 4963 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 196 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 3173 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 6642 . . . . . 6  |-  suc  On  =  On
1413unieqi 4260 . . . . 5  |-  U. suc  On  =  U. On
15 unon 6665 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2486 . . . 4  |-  U. suc  On  =  On
17 suceq 4952 . . . . 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 2524 . . 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 1395    e. wcel 1819   U.cuni 4251   Tr wtr 4550   Ord word 4886   Oncon0 4887   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893
This theorem is referenced by:  orduniss2  6667  onsucuni2  6668  nlimsucg  6676  tz7.44-2  7091  ttukeylem7  8912  tsksuc  9157  dfrdg2  29402  ontgsucval  30059  onsuctopon  30061  limsucncmpi  30072
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