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Theorem onintrab2im 4325
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 3104 . 2  |-  { x  e.  On  |  ph }  C_  On
2 nfrab1 2546 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
32nfcri 2222 . . . 4  |-  F/ x  y  e.  { x  e.  On  |  ph }
43nfex 1573 . . 3  |-  F/ x E. y  y  e.  { x  e.  On  |  ph }
5 rabid 2542 . . . . 5  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
6 elex2 2635 . . . . 5  |-  ( x  e.  { x  e.  On  |  ph }  ->  E. y  y  e. 
{ x  e.  On  |  ph } )
75, 6sylbir 133 . . . 4  |-  ( ( x  e.  On  /\  ph )  ->  E. y 
y  e.  { x  e.  On  |  ph }
)
87ex 113 . . 3  |-  ( x  e.  On  ->  ( ph  ->  E. y  y  e. 
{ x  e.  On  |  ph } ) )
94, 8rexlimi 2482 . 2  |-  ( E. x  e.  On  ph  ->  E. y  y  e. 
{ x  e.  On  |  ph } )
10 onintonm 4324 . 2  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
E. y  y  e. 
{ x  e.  On  |  ph } )  ->  |^| { x  e.  On  |  ph }  e.  On )
111, 9, 10sylancr 405 1  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438   E.wrex 2360   {crab 2363    C_ wss 2997   |^|cint 3683   Oncon0 4181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by:  cardcl  6788
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