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Theorem dfom5 8181
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 8179 . . 3  |-  ( y  e.  om  <->  A. x
( Lim  x  ->  y  e.  x ) )
2 vex 3060 . . . 4  |-  y  e. 
_V
32elintab 4259 . . 3  |-  ( y  e.  |^| { x  |  Lim  x }  <->  A. x
( Lim  x  ->  y  e.  x ) )
41, 3bitr4i 260 . 2  |-  ( y  e.  om  <->  y  e.  |^|
{ x  |  Lim  x } )
54eqriv 2459 1  |-  om  =  |^| { x  |  Lim  x }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453    = wceq 1455    e. wcel 1898   {cab 2448   |^|cint 4248   Lim wlim 5443   omcom 6719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610  ax-inf2 8172
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6720
This theorem is referenced by: (None)
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