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Theorem dfom5 7595
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 7593 . . 3  |-  ( y  e.  om  <->  A. x
( Lim  x  ->  y  e.  x ) )
2 vex 2951 . . . 4  |-  y  e. 
_V
32elintab 4053 . . 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 2432 1  |-  om  =  |^| { x  |  Lim  x }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2421   |^|cint 4042   Lim wlim 4574   omcom 4837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-inf2 7586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838
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