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Theorem unon 4265
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 3612 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4149 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2445 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 118 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2577 . . . . 5  |-  x  e. 
_V
65sucid 4182 . . . 4  |-  x  e. 
suc  x
7 suceloni 4255 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3613 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 399 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 121 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2053 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   E.wrex 2324   U.cuni 3608   Oncon0 4128   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by:  limon  4267  onintonm  4271
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