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Theorem onintrab2 4774
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 4351 . 2  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  _V )
2 onintrab 4773 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  _V  <->  |^| { x  e.  On  |  ph }  e.  On )
31, 2bitri 241 1  |-  ( E. x  e.  On  ph  <->  |^|
{ x  e.  On  |  ph }  e.  On )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948   |^|cint 4042   Oncon0 4573
This theorem is referenced by:  oeeulem  6836  cardmin2  7875  cardaleph  7960  cardmin  8429  nobndlem2  25613  nobndlem4  25615  nobndlem6  25617  nofulllem4  25625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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