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Theorem dfom5 7367
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 7365 . . 3  |-  ( y  e.  om  <->  A. x
( Lim  x  ->  y  e.  x ) )
2 vex 2804 . . . 4  |-  y  e. 
_V
32elintab 3889 . . 3  |-  ( y  e.  |^| { x  |  Lim  x }  <->  A. x
( Lim  x  ->  y  e.  x ) )
41, 3bitr4i 243 . 2  |-  ( y  e.  om  <->  y  e.  |^|
{ x  |  Lim  x } )
54eqriv 2293 1  |-  om  =  |^| { x  |  Lim  x }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   |^|cint 3878   Lim wlim 4409   omcom 4672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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