Theorem scottex 8196

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 4523	. . . 4 |- (/) e. _V
2		eleq1 2523	. . . 4 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 233	. . 3 |- ( A = (/) -> A e. _V )
4		rabexg 4543	. . 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 3748	. . 3 |- ( -. A = (/) <-> E. y y e. A )
7		nfra1 2806	. . . . . 6 |- F/ y A. y e. A ( rank ` x ) C_ ( rank ` y )
8		nfcv 2613	. . . . . 6 |- F/_ y A
9	7, 8	nfrab 3001	. . . . 5 |- F/_ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) }
10	9	nfel1 2628	. . . 4 |- F/ y { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } e. _V
11		rsp 2887	. . . . . . . 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 2826	. . . . . 6 |- ( y e. A -> A. x e. A ( A. y e. A ( rank ` x ) C_ ( rank ` y ) -> ( rank ` x ) C_ ( rank ` y ) ) )
14		ss2rab 3529	. . . . . 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 8106	. . . . . . . 8 |- ( rank ` y ) e. On
17		fveq2 5792	. . . . . . . . . . . 12 |- ( x = w -> ( rank ` x ) = ( rank ` w ) )
18	17	sseq1d 3484	. . . . . . . . . . 11 |- ( x = w -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` w ) C_ ( rank ` y ) ) )
19	18	elrab 3217	. . . . . . . . . 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 2892	. . . . . . . 8 |- A. w e. { x e. A | ( rank ` x ) C_ ( rank ` y ) } ( rank ` w ) C_ ( rank ` y )
22		sseq2 3479	. . . . . . . . . 10 |- ( z = ( rank ` y ) -> ( ( rank ` w ) C_ z <-> ( rank ` w ) C_ ( rank ` y ) ) )
23	22	ralbidv 2841	. . . . . . . . 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 3172	. . . . . . . 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 8152	. . . . . . 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 4537	. . . . 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 1847	. . 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 1370   E.wex 1587   e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071   C_ wss 3429   (/)c0 3738   Oncon0 4820   ` cfv 5519   rankcrnk 8074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-reg 7911  ax-inf2 7951

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-recs 6935  df-rdg 6969  df-r1 8075  df-rank 8076

This theorem is referenced by:  scottexs  8198  cplem2  8201  kardex  8205  scottexf  29121