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Theorem onintrab2 6619
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 4552 . 2  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  _V )
2 onintrab 6618 . 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 1842   E.wrex 2754   {crab 2757   _Vcvv 3058   |^|cint 4226   Oncon0 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412  df-on 5413
This theorem is referenced by:  oeeulem  7286  cardmin2  8410  cardaleph  8501  cardmin  8970  nobndlem2  30140  nobndlem4  30142  nobndlem6  30144  nofulllem4  30152
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