Theorem scottex 8084

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 4417	. . . 4 |- (/) e. _V
2		eleq1 2498	. . . 4 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 233	. . 3 |- ( A = (/) -> A e. _V )
4		rabexg 4437	. . 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 3642	. . 3 |- ( -. A = (/) <-> E. y y e. A )
7		nfra1 2761	. . . . . 6 |- F/ y A. y e. A ( rank ` x ) C_ ( rank ` y )
8		nfcv 2574	. . . . . 6 |- F/_ y A
9	7, 8	nfrab 2897	. . . . 5 |- F/_ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) }
10	9	nfel1 2584	. . . 4 |- F/ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V
11		rsp 2771	. . . . . . . 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 2795	. . . . . 6 |- ( y e. A -> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
14		ss2rab 3423	. . . . . 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 7994	. . . . . . . 8 |- ( rank ` y ) e. On
17		fveq2 5686	. . . . . . . . . . . 12 |- ( x = w -> ( rank ` x ) = ( rank ` w ) )
18	17	sseq1d 3378	. . . . . . . . . . 11 |- ( x = w -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` w ) C_ ( rank ` y ) ) )
19	18	elrab 3112	. . . . . . . . . 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 2776	. . . . . . . 8 |- A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y )
22		sseq2 3373	. . . . . . . . . 10 |- ( z = ( rank ` y ) -> ( ( rank ` w ) C_ z <-> ( rank ` w ) C_ ( rank ` y ) ) )
23	22	ralbidv 2730	. . . . . . . . 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 3068	. . . . . . . 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 8040	. . . . . . 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 4431	. . . . 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 1844	. . 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 1369   E.wex 1586   e. wcel 1756   A.wral 2710   E.wrex 2711   {crab 2714   _Vcvv 2967   C_ wss 3323   (/)c0 3632   Oncon0 4714   ` cfv 5413   rankcrnk 7962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-reg 7799  ax-inf2 7839

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964

This theorem is referenced by:  scottexs  8086  cplem2  8089  kardex  8093  scottexf  28933