Theorem scottex 3541

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) (_ (rank` y)} e. V

Distinct variable group(s):   x,y,A

Proof of Theorem scottex

Step	Hyp	Ref	Expression
1		0ex 1745	. . . 4 |- (/) e. V
2		eleq1 1149	. . . 4 |- (A = (/) -> (A e. V <-> (/) e. V))
3	1, 2	mpbiri 169	. . 3 |- (A = (/) -> A e. V)
4		rabexg 1705	. . 3 |- (A e. V -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
5	3, 4	syl 12	. 2 |- (A = (/) -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
6		n0 1714	. . 3 |- (-. A = (/) <-> E.y y e. A)
7		hbra1 1237	. . . . . 6 |- (A.y e. A (rank` x) (_ (rank` y) -> A.yA.y e. A (rank` x) (_ (rank` y))
8		ax-17 925	. . . . . 6 |- (z e. A -> A.y z e. A)
9	7, 8	hbrab 1311	. . . . 5 |- (z e. {x e. A | A.y e. A (rank` x) (_ (rank` y)} -> A.y z e. {x e. A | A.y e. A (rank` x) (_ (rank` y)})
10		ax-17 925	. . . . 5 |- (z e. V -> A.y z e. V)
11	9, 10	hbel 1172	. . . 4 |- ({x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V -> A.y{x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
12		ra4 1243	. . . . . . . . 9 |- (A.y e. A (rank` x) (_ (rank` y) -> (y e. A -> (rank` x) (_ (rank` y)))
13	12	com12 13	. . . . . . . 8 |- (y e. A -> (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y)))
14	13	a1d 14	. . . . . . 7 |- (y e. A -> (x e. A -> (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y))))
15	14	r19.21aiv 1259	. . . . . 6 |- (y e. A -> A.x e. A (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y)))
16		ss2rab 1553	. . . . . 6 |- ({x e. A | A.y e. A (rank` x) (_ (rank` y)} (_ {x e. A | (rank` x) (_ (rank` y)} <-> A.x e. A (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y)))
17	15, 16	sylibr 175	. . . . 5 |- (y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} (_ {x e. A | (rank` x) (_ (rank` y)})
18		rankon 3515	. . . . . . . 8 |- (rank` y) e. On
19		fveq2 2832	. . . . . . . . . . . 12 |- (x = w -> (rank` x) = (rank` w))
20	19	sseq1d 1527	. . . . . . . . . . 11 |- (x = w -> ((rank` x) (_ (rank` y) <-> (rank` w) (_ (rank` y)))
21	20	elrab 1422	. . . . . . . . . 10 |- (w e. {x e. A | (rank` x) (_ (rank` y)} <-> (w e. A /\ (rank` w) (_ (rank` y)))
22	21	pm3.27bd 263	. . . . . . . . 9 |- (w e. {x e. A | (rank` x) (_ (rank` y)} -> (rank` w) (_ (rank` y))
23	22	rgen 1247	. . . . . . . 8 |- A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ (rank` y)
24		sseq2 1522	. . . . . . . . . 10 |- (z = (rank` y) -> ((rank` w) (_ z <-> (rank` w) (_ (rank` y)))
25	24	biraldv 1219	. . . . . . . . 9 |- (z = (rank` y) -> (A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ z <-> A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ (rank` y)))
26	25	rcla4ev 1403	. . . . . . . 8 |- (((rank` y) e. On /\ A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ (rank` y)) -> E.z e. On A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ z)
27	18, 23, 26	mp2an 520	. . . . . . 7 |- E.z e. On A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ z
28		bndrank 3526	. . . . . . 7 |- (E.z e. On A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ z -> {x e. A | (rank` x) (_ (rank` y)} e. V)
29	27, 28	ax-mp 6	. . . . . 6 |- {x e. A | (rank` x) (_ (rank` y)} e. V
30	29	ssex 1700	. . . . 5 |- ({x e. A | A.y e. A (rank` x) (_ (rank` y)} (_ {x e. A | (rank` x) (_ (rank` y)} -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
31	17, 30	syl 12	. . . 4 |- (y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
32	11, 31	19.23ai 746	. . 3 |- (E.y y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
33	6, 32	sylbi 174	. 2 |- (-. A = (/) -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
34	5, 33	pm2.61i 110	1 |- {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V

Syntax hints:  -. wn 1   -> wi 2  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  {crab 1204  Vcvv 1348   (_ wss 1487  (/)c0 1707  Oncon0 2199  ` cfv 2422  rankcrnk 3486

This theorem is referenced by:  scottexs 3543  cplem2 3546  kardex 3550

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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