Theorem scottex 8748

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 4790	. . . 4 |- (/) e. _V
2		eleq1 2689	. . . 4 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 248	. . 3 |- ( A = (/) -> A e. _V )
4		rabexg 4812	. . 3 |- ( A e. _V -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
5	3, 4	syl 17	. 2 |- ( A = (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
6		neq0 3930	. . 3 |- ( -. A = (/) <-> E. y y e. A )
7		nfra1 2941	. . . . . 6 |- F/ y A. y e. A ( rank ` x ) C_ ( rank ` y )
8		nfcv 2764	. . . . . 6 |- F/_ y A
9	7, 8	nfrab 3123	. . . . 5 |- F/_ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) }
10	9	nfel1 2779	. . . 4 |- F/ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V
11		rsp 2929	. . . . . . . 8 |- ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( y e. A -> ( rank ` x ) C_ ( rank ` y ) ) )
12	11	com12 32	. . . . . . 7 |- ( y e. A -> ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
13	12	ralrimivw 2967	. . . . . 6 |- ( y e. A -> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
14		ss2rab 3678	. . . . . 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 224	. . . . 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 8658	. . . . . . . 8 |- ( rank ` y ) e. On
17		fveq2 6191	. . . . . . . . . . . 12 |- ( x = w -> ( rank ` x ) = ( rank ` w ) )
18	17	sseq1d 3632	. . . . . . . . . . 11 |- ( x = w -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` w ) C_ ( rank ` y ) ) )
19	18	elrab 3363	. . . . . . . . . 10 |- ( w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } <-> ( w e. A /\ ( rank ` w ) C_ ( rank ` y ) ) )
20	19	simprbi 480	. . . . . . . . 9 |- ( w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } -> ( rank ` w ) C_ ( rank ` y ) )
21	20	rgen 2922	. . . . . . . 8 |- A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y )
22		sseq2 3627	. . . . . . . . . 10 |- ( z = ( rank ` y ) -> ( ( rank ` w ) C_ z <-> ( rank ` w ) C_ ( rank ` y ) ) )
23	22	ralbidv 2986	. . . . . . . . 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 3309	. . . . . . . 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 708	. . . . . . 7 |- E. z e. On A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ z
26		bndrank 8704	. . . . . . 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 4802	. . . . 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 17	. . . 4 |- ( y e. A -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
30	10, 29	exlimi 2086	. . 3 |- ( E. y y e. A -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
31	6, 30	sylbi 207	. 2 |- ( -. A = (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V )
32	5, 31	pm2.61i 176	1 |- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   Oncon0 5723   ` cfv 5888   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  scottexs  8750  cplem2  8753  kardex  8757  scottexf  33976