Theorem onintrab2im 4244

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 3025	. 2 |- { x e. On | ph } C_ On
2		nfrab1 2489	. . . . 5 |- F/_ x { x e. On | ph }
3	2	nfcri 2172	. . . 4 |- F/ x y e. { x e. On | ph }
4	3	nfex 1528	. . 3 |- F/ x E. y y e. { x e. On | ph }
5		rabid 2485	. . . . 5 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
6		elex2 2570	. . . . 5 |- ( x e. { x e. On | ph } -> E. y y e. { x e. On | ph } )
7	5, 6	sylbir 125	. . . 4 |- ( ( x e. On /\ ph ) -> E. y y e. { x e. On | ph } )
8	7	ex 108	. . 3 |- ( x e. On -> ( ph -> E. y y e. { x e. On | ph } ) )
9	4, 8	rexlimi 2426	. 2 |- ( E. x e. On ph -> E. y y e. { x e. On | ph } )
10		onintonm 4243	. 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 393	1 |- ( E. x e. On ph -> |^| { x e. On | ph } e. On )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   e. wcel 1393   E.wrex 2307   {crab 2310   C_ wss 2917   |^|cint 3615   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by:  cardcl  6361