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Theorem onintrab2 6615
Description: An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
onintrab2  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  On )

Proof of Theorem onintrab2
StepHypRef Expression
1 intexrab 4606 . 2  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  _V )
2 onintrab 6614 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  _V  <->  |^| { x  e.  On  |  ph }  e.  On )
31, 2bitri 249 1  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  On )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   |^|cint 4282   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  oeeulem  7247  cardmin2  8375  cardaleph  8466  cardmin  8935  nobndlem2  29030  nobndlem4  29032  nobndlem6  29034  nofulllem4  29042
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