Theorem scottex 8299

Description: Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)

Assertion

Ref	Expression
scottex	|- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V

Distinct variable group:   x, y, A

Proof of Theorem scottex

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4577	. . . 4 |- (/) e. _V
2		eleq1 2539	. . . 4 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 233	. . 3 |- ( A = (/) -> A e. _V )
4		rabexg 4597	. . 3 |- ( A e. _V -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
5	3, 4	syl 16	. 2 |- ( A = (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
6		neq0 3795	. . 3 |- ( -. A = (/) <-> E. y y e. A )
7		nfra1 2845	. . . . . 6 |- F/ y A. y e. A ( rank ` x ) C_ ( rank ` y )
8		nfcv 2629	. . . . . 6 |- F/_ y A
9	7, 8	nfrab 3043	. . . . 5 |- F/_ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) }
10	9	nfel1 2645	. . . 4 |- F/ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V
11		rsp 2830	. . . . . . . 8 |- ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( y e. A -> ( rank ` x ) C_ ( rank ` y ) ) )
12	11	com12 31	. . . . . . 7 |- ( y e. A -> ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
13	12	ralrimivw 2879	. . . . . 6 |- ( y e. A -> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
14		ss2rab 3576	. . . . . 6 |- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } C_ { x e. A | ( rank ` x ) C_ ( rank ` y ) } <-> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
15	13, 14	sylibr 212	. . . . 5 |- ( y e. A -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } C_ { x e. A | ( rank ` x ) C_ ( rank ` y ) } )
16		rankon 8209	. . . . . . . 8 |- ( rank ` y ) e. On
17		fveq2 5864	. . . . . . . . . . . 12 |- ( x = w -> ( rank ` x ) = ( rank ` w ) )
18	17	sseq1d 3531	. . . . . . . . . . 11 |- ( x = w -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` w ) C_ ( rank ` y ) ) )
19	18	elrab 3261	. . . . . . . . . 10 |- ( w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } <-> ( w e. A /\ ( rank ` w ) C_ ( rank ` y ) ) )
20	19	simprbi 464	. . . . . . . . 9 |- ( w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } -> ( rank ` w ) C_ ( rank ` y ) )
21	20	rgen 2824	. . . . . . . 8 |- A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y )
22		sseq2 3526	. . . . . . . . . 10 |- ( z = ( rank ` y ) -> ( ( rank ` w ) C_ z <-> ( rank ` w ) C_ ( rank ` y ) ) )
23	22	ralbidv 2903	. . . . . . . . 9 |- ( z = ( rank ` y ) -> ( A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ z <-> A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y ) ) )
24	23	rspcev 3214	. . . . . . . 8 |- ( ( ( rank ` y ) e. On /\ A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y ) ) -> E. z e. On A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ z )
25	16, 21, 24	mp2an 672	. . . . . . 7 |- E. z e. On A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ z
26		bndrank 8255	. . . . . . 7 |- ( E. z e. On A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ z -> { x e. A | ( rank ` x ) C_ ( rank ` y ) } e. _V )
27	25, 26	ax-mp 5	. . . . . 6 |- { x e. A | ( rank ` x ) C_ ( rank ` y ) } e. _V
28	27	ssex 4591	. . . . 5 |- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } C_ { x e. A | ( rank ` x ) C_ ( rank ` y ) } -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
29	15, 28	syl 16	. . . 4 |- ( y e. A -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
30	10, 29	exlimi 1859	. . 3 |- ( E. y y e. A -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
31	6, 30	sylbi 195	. 2 |- ( -. A = (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
32	5, 31	pm2.61i 164	1 |- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V

Syntax hints:   -. wn 3   -> wi 4   = wceq 1379   E.wex 1596   e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   C_ wss 3476   (/)c0 3785   Oncon0 4878   ` cfv 5586   rankcrnk 8177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073  df-r1 8178  df-rank 8179

This theorem is referenced by:  scottexs  8301  cplem2  8304  kardex  8308  scottexf  30180