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Theorem unon 4326
Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
Assertion
Ref Expression
unon  |-  U. On  =  On

Proof of Theorem unon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3655 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4209 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2484 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 119 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2622 . . . . 5  |-  x  e. 
_V
65sucid 4242 . . . 4  |-  x  e. 
suc  x
7 suceloni 4316 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3656 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 405 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 124 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2085 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   E.wrex 2360   U.cuni 3651   Oncon0 4188   suc csuc 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-iord 4191  df-on 4193  df-suc 4196
This theorem is referenced by:  limon  4328  onintonm  4332  tfri1dALT  6108  rdgon  6143
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