Theorem onintrab2im 4495

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 3227	. 2 |- { x e. On | ph } C_ On
2		nfrab1 2645	. . . . 5 |- F/_ x { x e. On | ph }
3	2	nfcri 2302	. . . 4 |- F/ x y e. { x e. On | ph }
4	3	nfex 1625	. . 3 |- F/ x E. y y e. { x e. On | ph }
5		rabid 2641	. . . . 5 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
6		elex2 2742	. . . . 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 2576	. 2 |- ( E. x e. On ph -> E. y y e. { x e. On | ph } )
10		onintonm 4494	. 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 411	1 |- ( E. x e. On ph -> |^| { x e. On | ph } e. On )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   E.wrex 2445   {crab 2448   C_ wss 3116   |^|cint 3824   Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  cardcl  7137