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Theorem unon 4621
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem unon
StepHypRef Expression
1 eluni2 3832 . . . 4  |-  ( x  e.  U. On  <->  E. y  e.  On  x  e.  y )
2 onelon 4416 . . . . 5  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
32rexlimiva 2663 . . . 4  |-  ( E. y  e.  On  x  e.  y  ->  x  e.  On )
41, 3sylbi 189 . . 3  |-  ( x  e.  U. On  ->  x  e.  On )
5 vex 2792 . . . . 5  |-  x  e. 
_V
65sucid 4470 . . . 4  |-  x  e. 
suc  x
7 suceloni 4603 . . . 4  |-  ( x  e.  On  ->  suc  x  e.  On )
8 elunii 3833 . . . 4  |-  ( ( x  e.  suc  x  /\  suc  x  e.  On )  ->  x  e.  U. On )
96, 7, 8sylancr 646 . . 3  |-  ( x  e.  On  ->  x  e.  U. On )
104, 9impbii 182 . 2  |-  ( x  e.  U. On  <->  x  e.  On )
1110eqriv 2281 1  |-  U. On  =  On
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685   E.wrex 2545   U.cuni 3828   Oncon0 4391   suc csuc 4393
This theorem is referenced by:  ordunisuc  4622  limon  4626  orduninsuc  4633  ordtoplem  24281  ordcmp  24293
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397
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