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Theorem onintrab 4629
Description: The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
onintrab  |-  ( |^| { x  e.  On  |  ph }  e.  _V  <->  |^| { x  e.  On  |  ph }  e.  On )

Proof of Theorem onintrab
StepHypRef Expression
1 intex 4204 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  |^| { x  e.  On  |  ph }  e.  _V )
2 ssrab2 3292 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 oninton 4628 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  On )
42, 3mpan 651 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  On )
51, 4sylbir 204 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  _V  ->  |^|
{ x  e.  On  |  ph }  e.  On )
6 elex 2830 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  On  ->  |^|
{ x  e.  On  |  ph }  e.  _V )
75, 6impbii 180 1  |-  ( |^| { x  e.  On  |  ph }  e.  _V  <->  |^| { x  e.  On  |  ph }  e.  On )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1701    =/= wne 2479   {crab 2581   _Vcvv 2822    C_ wss 3186   (/)c0 3489   |^|cint 3899   Oncon0 4429
This theorem is referenced by:  onintrab2  4630  sltval2  24695  sltres  24703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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