ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unon Unicode version

Theorem unon 4341
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 3663 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4220 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2485 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 120 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2623 . . . . 5  |-  x  e. 
_V
65sucid 4253 . . . 4  |-  x  e. 
suc  x
7 suceloni 4331 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3664 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 406 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 125 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2086 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1290    e. wcel 1439   E.wrex 2361   U.cuni 3659   Oncon0 4199   suc csuc 4201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207
This theorem is referenced by:  limon  4343  onintonm  4347  tfri1dALT  6130  rdgon  6165
  Copyright terms: Public domain W3C validator