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

Theorem ordunisuc 4841
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 4798 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 suceq 4675 . . . . . 6  |-  ( x  =  A  ->  suc  x  =  suc  A )
32unieqd 4050 . . . . 5  |-  ( x  =  A  ->  U. suc  x  =  U. suc  A
)
4 id 21 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4eqeq12d 2456 . . . 4  |-  ( x  =  A  ->  ( U. suc  x  =  x  <->  U. suc  A  =  A ) )
6 eloni 4620 . . . . . 6  |-  ( x  e.  On  ->  Ord  x )
7 ordtr 4624 . . . . . 6  |-  ( Ord  x  ->  Tr  x
)
86, 7syl 16 . . . . 5  |-  ( x  e.  On  ->  Tr  x )
9 vex 2965 . . . . . 6  |-  x  e. 
_V
109unisuc 4686 . . . . 5  |-  ( Tr  x  <->  U. suc  x  =  x )
118, 10sylib 190 . . . 4  |-  ( x  e.  On  ->  U. suc  x  =  x )
125, 11vtoclga 3023 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
13 sucon 4817 . . . . . 6  |-  suc  On  =  On
1413unieqi 4049 . . . . 5  |-  U. suc  On  =  U. On
15 unon 4840 . . . . 5  |-  U. On  =  On
1614, 15eqtri 2462 . . . 4  |-  U. suc  On  =  On
17 suceq 4675 . . . . 5  |-  ( A  =  On  ->  suc  A  =  suc  On )
1817unieqd 4050 . . . 4  |-  ( A  =  On  ->  U. suc  A  =  U. suc  On )
19 id 21 . . . 4  |-  ( A  =  On  ->  A  =  On )
2016, 18, 193eqtr4a 2500 . . 3  |-  ( A  =  On  ->  U. suc  A  =  A )
2112, 20jaoi 370 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  U. suc  A  =  A )
221, 21sylbi 189 1  |-  ( Ord 
A  ->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    = wceq 1653    e. wcel 1727   U.cuni 4039   Tr wtr 4327   Ord word 4609   Oncon0 4610   suc csuc 4612
This theorem is referenced by:  orduniss2  4842  onsucuni2  4843  nlimsucg  4851  tz7.44-2  6694  ttukeylem7  8426  tsksuc  8668  dfrdg2  25454  ontgsucval  26213  onsuctopon  26215  limsucncmpi  26226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-suc 4616
  Copyright terms: Public domain W3C validator