MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordunisuc Unicode version

Theorem ordunisuc 4623
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 4580 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4457 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 3838 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4402 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4406 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 15 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 2791 . . . . . 6  |-  x  e. 
_V
109unisuc 4468 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 188 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 2849 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 4599 . . . . . 6  |-  suc  On  =  On
1413unieqi 3837 . . . . 5  |-  U. suc  On  =  U. On
15 unon 4622 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2303 . . . 4  |-  U. suc  On  =  On
17 suceq 4457 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 3838 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 19 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2341 . . 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 1623    e. wcel 1684   U.cuni 3827   Tr wtr 4113   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  orduniss2  4624  onsucuni2  4625  nlimsucg  4633  tz7.44-2  6420  ttukeylem7  8142  tsksuc  8384  dfrdg2  24152  ontgsucval  24871  onsuctopon  24873  limsucncmpi  24884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
  Copyright terms: Public domain W3C validator