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Theorem ordunisuc 6556
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 6513 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4895 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 4212 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2476 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4840 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4844 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 16 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 3081 . . . . . 6  |-  x  e. 
_V
109unisuc 4906 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 196 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 3142 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 6532 . . . . . 6  |-  suc  On  =  On
1413unieqi 4211 . . . . 5  |-  U. suc  On  =  U. On
15 unon 6555 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2483 . . . 4  |-  U. suc  On  =  On
17 suceq 4895 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 4212 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 22 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2521 . . 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 1370    e. wcel 1758   U.cuni 4202   Tr wtr 4496   Ord word 4829   Oncon0 4830   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836
This theorem is referenced by:  orduniss2  6557  onsucuni2  6558  nlimsucg  6566  tz7.44-2  6976  ttukeylem7  8799  tsksuc  9044  dfrdg2  27776  ontgsucval  28445  onsuctopon  28447  limsucncmpi  28458
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