Theorem scottex 4699

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:   x,y,A

Proof of Theorem scottex

Step	Hyp	Ref	Expression
1		0ex 2707	. . . 4 |- (/) e. V
2		eleq1 1532	. . . 4 |- (A = (/) -> (A e. V <-> (/) e. V))
3	1, 2	mpbiri 194	. . 3 |- (A = (/) -> A e. V)
4		rabexg 2720	. . 3 |- (A e. V -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
5	3, 4	syl 10	. 2 |- (A = (/) -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
6		n0 2286	. . 3 |- (-. A = (/) <-> E.y y e. A)
7		hbra1 1685	. . . . . 6 |- (A.y e. A (rank` x) (_ (rank` y) -> A.yA.y e. A (rank` x) (_ (rank` y))
8		ax-17 970	. . . . . 6 |- (z e. A -> A.y z e. A)
9	7, 8	hbrab 1771	. . . . 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-17 970	. . . . 5 |- (z e. V -> A.y z e. V)
11	9, 10	hbel 1564	. . . 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 1692	. . . . . . . . 9 |- (A.y e. A (rank` x) (_ (rank` y) -> (y e. A -> (rank` x) (_ (rank` y)))
13	12	com12 11	. . . . . . . 8 |- (y e. A -> (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y)))
14	13	a1d 12	. . . . . . 7 |- (y e. A -> (x e. A -> (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y))))
15	14	r19.21aiv 1711	. . . . . 6 |- (y e. A -> A.x e. A (A.y e. A (rank` x) (_ (rank` y) -> (rank` x) (_ (rank` y)))
16		ss2rab 2120	. . . . . 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 200	. . . . 5 |- (y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} (_ {x e. A | (rank` x) (_ (rank` y)})
18		rankon 4654	. . . . . . . 8 |- (rank` y) e. On
19		fveq2 3719	. . . . . . . . . . . 12 |- (x = w -> (rank` x) = (rank` w))
20	19	sseq1d 2085	. . . . . . . . . . 11 |- (x = w -> ((rank` x) (_ (rank` y) <-> (rank` w) (_ (rank` y)))
21	20	elrab 1902	. . . . . . . . . 10 |- (w e. {x e. A | (rank` x) (_ (rank` y)} <-> (w e. A /\ (rank` w) (_ (rank` y)))
22	21	pm3.27bi 326	. . . . . . . . 9 |- (w e. {x e. A | (rank` x) (_ (rank` y)} -> (rank` w) (_ (rank` y))
23	22	rgen 1696	. . . . . . . 8 |- A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ (rank` y)
24		sseq2 2080	. . . . . . . . . 10 |- (z = (rank` y) -> ((rank` w) (_ z <-> (rank` w) (_ (rank` y)))
25	24	ralbidv 1661	. . . . . . . . 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 1874	. . . . . . . 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 696	. . . . . . 7 |- E.z e. On A.w e. {x e. A | (rank` x) (_ (rank` y)} (rank` w) (_ z
28		bndrank 4665	. . . . . . 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 7	. . . . . 6 |- {x e. A | (rank` x) (_ (rank` y)} e. V
30	29	ssex 2715	. . . . 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 10	. . . 4 |- (y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
32	11, 31	19.23ai 1063	. . 3 |- (E.y y e. A -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
33	6, 32	sylbi 199	. 2 |- (-. A = (/) -> {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V)
34	5, 33	pm2.61i 126	1 |- {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V

Syntax hints:  -. wn 2   -> wi 3   = wceq 955   e. wcel 957  E.wex 979  A.wral 1643  E.wrex 1644  {crab 1646  Vcvv 1808   (_ wss 2044  (/)c0 2277  Oncon0 2944  ` cfv 3178  rankcrnk 4625

This theorem is referenced by:  scottexs 4701  cplem2 4704  kardex 4708

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-r1 4626  df-rank 4627

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