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Theorem onintrab2im 4404
Description: An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
onintrab2im  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )

Proof of Theorem onintrab2im
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3152 . 2  |-  { x  e.  On  |  ph }  C_  On
2 nfrab1 2587 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
32nfcri 2252 . . . 4  |-  F/ x  y  e.  { x  e.  On  |  ph }
43nfex 1601 . . 3  |-  F/ x E. y  y  e.  { x  e.  On  |  ph }
5 rabid 2583 . . . . 5  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
6 elex2 2676 . . . . 5  |-  ( x  e.  { x  e.  On  |  ph }  ->  E. y  y  e. 
{ x  e.  On  |  ph } )
75, 6sylbir 134 . . . 4  |-  ( ( x  e.  On  /\  ph )  ->  E. y 
y  e.  { x  e.  On  |  ph }
)
87ex 114 . . 3  |-  ( x  e.  On  ->  ( ph  ->  E. y  y  e. 
{ x  e.  On  |  ph } ) )
94, 8rexlimi 2519 . 2  |-  ( E. x  e.  On  ph  ->  E. y  y  e. 
{ x  e.  On  |  ph } )
10 onintonm 4403 . 2  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
E. y  y  e. 
{ x  e.  On  |  ph } )  ->  |^| { x  e.  On  |  ph }  e.  On )
111, 9, 10sylancr 410 1  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1453    e. wcel 1465   E.wrex 2394   {crab 2397    C_ wss 3041   |^|cint 3741   Oncon0 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by:  cardcl  7005
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