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Theorem onintrab2 6518
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 4554 . 2  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  _V )
2 onintrab 6517 . 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 1758   E.wrex 2797   {crab 2800   _Vcvv 3072   |^|cint 4231   Oncon0 4822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826
This theorem is referenced by:  oeeulem  7145  cardmin2  8274  cardaleph  8365  cardmin  8834  nobndlem2  27973  nobndlem4  27975  nobndlem6  27977  nofulllem4  27985
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