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Theorem dfom5 7540
Description:  om is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
dfom5  |-  om  =  |^| { x  |  Lim  x }

Proof of Theorem dfom5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elom3 7538 . . 3  |-  ( y  e.  om  <->  A. x
( Lim  x  ->  y  e.  x ) )
2 vex 2904 . . . 4  |-  y  e. 
_V
32elintab 4005 . . 3  |-  ( y  e.  |^| { x  |  Lim  x }  <->  A. x
( Lim  x  ->  y  e.  x ) )
41, 3bitr4i 244 . 2  |-  ( y  e.  om  <->  y  e.  |^|
{ x  |  Lim  x } )
54eqriv 2386 1  |-  om  =  |^| { x  |  Lim  x }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2375   |^|cint 3994   Lim wlim 4525   omcom 4787
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643  ax-inf2 7531
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-tr 4246  df-eprel 4437  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788
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