Theorem scottex 8320

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 4587	. . . 4 |- (/) e. _V
2		eleq1 2529	. . . 4 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 233	. . 3 |- ( A = (/) -> A e. _V )
4		rabexg 4606	. . 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 3804	. . 3 |- ( -. A = (/) <-> E. y y e. A )
7		nfra1 2838	. . . . . 6 |- F/ y A. y e. A ( rank ` x ) C_ ( rank ` y )
8		nfcv 2619	. . . . . 6 |- F/_ y A
9	7, 8	nfrab 3039	. . . . 5 |- F/_ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) }
10	9	nfel1 2635	. . . 4 |- F/ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V
11		rsp 2823	. . . . . . . 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 2872	. . . . . 6 |- ( y e. A -> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
14		ss2rab 3572	. . . . . 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 8230	. . . . . . . 8 |- ( rank ` y ) e. On
17		fveq2 5872	. . . . . . . . . . . 12 |- ( x = w -> ( rank ` x ) = ( rank ` w ) )
18	17	sseq1d 3526	. . . . . . . . . . 11 |- ( x = w -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` w ) C_ ( rank ` y ) ) )
19	18	elrab 3257	. . . . . . . . . 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 2817	. . . . . . . 8 |- A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y )
22		sseq2 3521	. . . . . . . . . 10 |- ( z = ( rank ` y ) -> ( ( rank ` w ) C_ z <-> ( rank ` w ) C_ ( rank ` y ) ) )
23	22	ralbidv 2896	. . . . . . . . 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 3210	. . . . . . . 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 8276	. . . . . . 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 4600	. . . . 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 1913	. . 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 1395   E.wex 1613   e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   C_ wss 3471   (/)c0 3793   Oncon0 4887   ` cfv 5594   rankcrnk 8198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200

This theorem is referenced by:  scottexs  8322  cplem2  8325  kardex  8329  scottexf  30738