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.

Step	Hyp	Ref	Expression
1		ssrab2 3080	. 2 |- { x e. On | ph } C_ On
2		nfrab1 2534	. . . . 5 |- F/_ x { x e. On | ph }
3	2	nfcri 2214	. . . 4 |- F/ x y e. { x e. On | ph }
4	3	nfex 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 } )
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 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 )
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 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