Theorem onintrab2im 4434

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.

Step	Hyp	Ref	Expression
1		ssrab2 3182	. 2 |- { x e. On | ph } C_ On
2		nfrab1 2610	. . . . 5 |- F/_ x { x e. On | ph }
3	2	nfcri 2275	. . . 4 |- F/ x y e. { x e. On | ph }
4	3	nfex 1616	. . 3 |- F/ x E. y y e. { x e. On | ph }
5		rabid 2606	. . . . 5 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
6		elex2 2702	. . . . 5 |- ( x e. { x e. On | ph } -> E. y y e. { x e. On | ph } )
7	5, 6	sylbir 134	. . . 4 |- ( ( x e. On /\ ph ) -> E. y y e. { x e. On | ph } )
8	7	ex 114	. . 3 |- ( x e. On -> ( ph -> E. y y e. { x e. On | ph } ) )
9	4, 8	rexlimi 2542	. 2 |- ( E. x e. On ph -> E. y y e. { x e. On | ph } )
10		onintonm 4433	. 2 |- ( ( { x e. On | ph } C_ On /\ E. y y e. { x e. On | ph } ) -> |^| { x e. On | ph } e. On )
11	1, 9, 10	sylancr 410	1 |- ( E. x e. On ph -> |^| { x e. On | ph } e. On )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1468   e. wcel 1480   E.wrex 2417   {crab 2420   C_ wss 3071   |^|cint 3771   Oncon0 4285

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293

This theorem is referenced by:  cardcl  7037