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Theorem onintrab2im 4270
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 3080 . 2  |-  { x  e.  On  |  ph }  C_  On
2 nfrab1 2534 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
32nfcri 2214 . . . 4  |-  F/ x  y  e.  { x  e.  On  |  ph }
43nfex 1569 . . 3  |-  F/ x E. y  y  e.  { x  e.  On  |  ph }
5 rabid 2530 . . . . 5  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
6 elex2 2616 . . . . 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 2471 . 2  |-  ( E. x  e.  On  ph  ->  E. y  y  e. 
{ x  e.  On  |  ph } )
10 onintonm 4269 . 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 1422    e. wcel 1434   E.wrex 2350   {crab 2353    C_ wss 2974   |^|cint 3644   Oncon0 4126
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134
This theorem is referenced by:  cardcl  6509
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