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Theorem unon 4770
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 3979 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4566 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2785 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 188 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2919 . . . . 5  |-  x  e. 
_V
65sucid 4620 . . . 4  |-  x  e. 
suc  x
7 suceloni 4752 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3980 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 645 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 181 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2401 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   E.wrex 2667   U.cuni 3975   Oncon0 4541   suc csuc 4543
This theorem is referenced by:  ordunisuc  4771  limon  4775  orduninsuc  4782  ordtoplem  26089  ordcmp  26101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547
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