Theorem onintrab2im 4262

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 3079	. 2 |- { x e. On | ph } C_ On
2		nfrab1 2533	. . . . 5 |- F/_ x { x e. On | ph }
3	2	nfcri 2213	. . . 4 |- F/ x y e. { x e. On | ph }
4	3	nfex 1568	. . 3 |- F/ x E. y y e. { x e. On | ph }
5		rabid 2529	. . . . 5 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
6		elex2 2615	. . . . 5 |- ( x e. { x e. On | ph } -> E. y y e. { x e. On | ph } )
7	5, 6	sylbir 133	. . . 4 |- ( ( x e. On /\ ph ) -> E. y y e. { x e. On | ph } )
8	7	ex 113	. . 3 |- ( x e. On -> ( ph -> E. y y e. { x e. On | ph } ) )
9	4, 8	rexlimi 2470	. 2 |- ( E. x e. On ph -> E. y y e. { x e. On | ph } )
10		onintonm 4261	. 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 405	1 |- ( E. x e. On ph -> |^| { x e. On | ph } e. On )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   e. wcel 1433   E.wrex 2349   {crab 2352   C_ wss 2973   |^|cint 3636   Oncon0 4118

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by:  cardcl  6450