Theorem scottex 5846

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

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

Step	Hyp	Ref	Expression
1		0ex 3446	. . . 4 |- (/) e. _V
2		eleq1 1957	. . . 4 |- (A = (/) -> (A e. _V <-> (/) e. _V))
3	1, 2	mpbiri 211	. . 3 |- (A = (/) -> A e. _V)
4		rabexg 3460	. . 3 |- (A e. _V -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
5	3, 4	syl 12	. 2 |- (A = (/) -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
6		neq0 2885	. . 3 |- (-. A = (/) <-> E.y y e. A)
7		hbra1 2147	. . . . . 6 |- (A.y e. A (rank` x) C_ (rank` y) -> A.yA.y e. A (rank` x) C_ (rank` y))
8		ax-17 1317	. . . . . 6 |- (z e. A -> A.y z e. A)
9	7, 8	hbrab 2258	. . . . 5 |- (z e. {x e. A | A.y e. A (rank` x) C_ (rank` y)} -> A.y z e. {x e. A | A.y e. A (rank` x) C_ (rank` y)})
10		ax-17 1317	. . . . 5 |- (z e. _V -> A.y z e. _V)
11	9, 10	hbel 1996	. . . 4 |- ({x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V -> A.y{x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
12		ra4 2155	. . . . . . . . 9 |- (A.y e. A (rank` x) C_ (rank` y) -> (y e. A -> (rank` x) C_ (rank` y)))
13	12	com12 14	. . . . . . . 8 |- (y e. A -> (A.y e. A (rank` x) C_ (rank` y) -> (rank` x) C_ (rank` y)))
14	13	a1d 15	. . . . . . 7 |- (y e. A -> (x e. A -> (A.y e. A (rank` x) C_ (rank` y) -> (rank` x) C_ (rank` y))))
15	14	r19.21aiv 2175	. . . . . 6 |- (y e. A -> A.x e. A (A.y e. A (rank` x) C_ (rank` y) -> (rank` x) C_ (rank` y)))
16		ss2rab 2683	. . . . . 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)))
17	15, 16	sylibr 217	. . . . 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)})
18		rankon 5782	. . . . . . . 8 |- (rank` y) e. On
19		fveq2 4681	. . . . . . . . . . . 12 |- (x = w -> (rank` x) = (rank` w))
20	19	sseq1d 2644	. . . . . . . . . . 11 |- (x = w -> ((rank` x) C_ (rank` y) <-> (rank` w) C_ (rank` y)))
21	20	elrab 2414	. . . . . . . . . 10 |- (w e. {x e. A | (rank` x) C_ (rank` y)} <-> (w e. A /\ (rank` w) C_ (rank` y)))
22	21	simprbi 353	. . . . . . . . 9 |- (w e. {x e. A | (rank` x) C_ (rank` y)} -> (rank` w) C_ (rank` y))
23	22	rgen 2159	. . . . . . . 8 |- A.w e. {x e. A | (rank` x) C_ (rank` y)} (rank` w) C_ (rank` y)
24		sseq2 2639	. . . . . . . . . 10 |- (z = (rank` y) -> ((rank` w) C_ z <-> (rank` w) C_ (rank` y)))
25	24	ralbidv 2123	. . . . . . . . 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)))
26	25	rcla4ev 2381	. . . . . . . 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)
27	18, 23, 26	mp2an 761	. . . . . . 7 |- E.z e. On A.w e. {x e. A | (rank` x) C_ (rank` y)} (rank` w) C_ z
28		bndrank 5793	. . . . . . 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)
29	27, 28	ax-mp 7	. . . . . 6 |- {x e. A | (rank` x) C_ (rank` y)} e. _V
30	29	ssex 3455	. . . . 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)
31	17, 30	syl 12	. . . 4 |- (y e. A -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
32	11, 31	19.23ai 1412	. . 3 |- (E.y y e. A -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
33	6, 32	sylbi 216	. 2 |- (-. A = (/) -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V)
34	5, 33	pm2.61i 140	1 |- {x e. A | A.y e. A (rank` x) C_ (rank` y)} e. _V

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  Oncon0 3657  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  scottexs 5848  cplem2 5851  kardex 5855

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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