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Theorem unon 4701
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 3910 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4496 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2738 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 187 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2867 . . . . 5  |-  x  e. 
_V
65sucid 4550 . . . 4  |-  x  e. 
suc  x
7 suceloni 4683 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3911 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 644 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 180 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2355 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   E.wrex 2620   U.cuni 3906   Oncon0 4471   suc csuc 4473
This theorem is referenced by:  ordunisuc  4702  limon  4706  orduninsuc  4713  ordtoplem  25433  ordcmp  25445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477
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