Theorem scott0 5847

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, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. A is empty).

Assertion

Ref	Expression
scott0	|- (A = (/) <-> {x e. A | A.y e. A (rank` x) C_ (rank` y)} = (/))

Distinct variable group:   x,y,A

Proof of Theorem scott0

Step	Hyp	Ref	Expression
1		rabeq 2289	. . 3 |- (A = (/) -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} = {x e. (/) | A.y e. A (rank` x) C_ (rank` y)})
2		rab0 2894	. . 3 |- {x e. (/) | A.y e. A (rank` x) C_ (rank` y)} = (/)
3	1, 2	syl6eq 1944	. 2 |- (A = (/) -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} = (/))
4		n0 2884	. . . . . . . . 9 |- (A =/= (/) <-> E.x x e. A)
5		hbre1 2150	. . . . . . . . . 10 |- (E.x e. A (rank` x) = (rank` x) -> A.xE.x e. A (rank` x) = (rank` x))
6		eqid 1884	. . . . . . . . . . 11 |- (rank` x) = (rank` x)
7		ra4e 2156	. . . . . . . . . . 11 |- ((x e. A /\ (rank` x) = (rank` x)) -> E.x e. A (rank` x) = (rank` x))
8	6, 7	mpan2 760	. . . . . . . . . 10 |- (x e. A -> E.x e. A (rank` x) = (rank` x))
9	5, 8	19.23ai 1412	. . . . . . . . 9 |- (E.x x e. A -> E.x e. A (rank` x) = (rank` x))
10	4, 9	sylbi 216	. . . . . . . 8 |- (A =/= (/) -> E.x e. A (rank` x) = (rank` x))
11		fvex 4689	. . . . . . . . . . . 12 |- (rank` x) e. _V
12		eqeq1 1890	. . . . . . . . . . . . 13 |- (y = (rank` x) -> (y = (rank` x) <-> (rank` x) = (rank` x)))
13	12	anbi2d 678	. . . . . . . . . . . 12 |- (y = (rank` x) -> ((x e. A /\ y = (rank` x)) <-> (x e. A /\ (rank` x) = (rank` x))))
14	11, 13	cla4ev 2371	. . . . . . . . . . 11 |- ((x e. A /\ (rank` x) = (rank` x)) -> E.y(x e. A /\ y = (rank` x)))
15	14	eximi 1387	. . . . . . . . . 10 |- (E.x(x e. A /\ (rank` x) = (rank` x)) -> E.xE.y(x e. A /\ y = (rank` x)))
16		excom 1393	. . . . . . . . . 10 |- (E.yE.x(x e. A /\ y = (rank` x)) <-> E.xE.y(x e. A /\ y = (rank` x)))
17	15, 16	sylibr 217	. . . . . . . . 9 |- (E.x(x e. A /\ (rank` x) = (rank` x)) -> E.yE.x(x e. A /\ y = (rank` x)))
18		df-rex 2110	. . . . . . . . 9 |- (E.x e. A (rank` x) = (rank` x) <-> E.x(x e. A /\ (rank` x) = (rank` x)))
19		df-rex 2110	. . . . . . . . . 10 |- (E.x e. A y = (rank` x) <-> E.x(x e. A /\ y = (rank` x)))
20	19	exbii 1398	. . . . . . . . 9 |- (E.yE.x e. A y = (rank` x) <-> E.yE.x(x e. A /\ y = (rank` x)))
21	17, 18, 20	3imtr4i 236	. . . . . . . 8 |- (E.x e. A (rank` x) = (rank` x) -> E.yE.x e. A y = (rank` x))
22	10, 21	syl 12	. . . . . . 7 |- (A =/= (/) -> E.yE.x e. A y = (rank` x))
23		abn0 2892	. . . . . . 7 |- ({y | E.x e. A y = (rank` x)} =/= (/) <-> E.yE.x e. A y = (rank` x))
24	22, 23	sylibr 217	. . . . . 6 |- (A =/= (/) -> {y | E.x e. A y = (rank` x)} =/= (/))
25		hbab1 1874	. . . . . . . . . 10 |- (z e. {y | E.x e. A y = (rank` x)} -> A.y z e. {y | E.x e. A y = (rank` x)})
26		ax-17 1317	. . . . . . . . . 10 |- (z e. On -> A.y z e. On)
27	25, 26	dfss2f 2612	. . . . . . . . 9 |- ({y | E.x e. A y = (rank` x)} C_ On <-> A.y(y e. {y | E.x e. A y = (rank` x)} -> y e. On))
28		abid 1873	. . . . . . . . . 10 |- (y e. {y | E.x e. A y = (rank` x)} <-> E.x e. A y = (rank` x))
29		rankon 5782	. . . . . . . . . . . . 13 |- (rank` x) e. On
30		eleq1 1957	. . . . . . . . . . . . 13 |- (y = (rank` x) -> (y e. On <-> (rank` x) e. On))
31	29, 30	mpbiri 211	. . . . . . . . . . . 12 |- (y = (rank` x) -> y e. On)
32	31	a1i 8	. . . . . . . . . . 11 |- (x e. A -> (y = (rank` x) -> y e. On))
33	32	r19.23aiv 2211	. . . . . . . . . 10 |- (E.x e. A y = (rank` x) -> y e. On)
34	28, 33	sylbi 216	. . . . . . . . 9 |- (y e. {y | E.x e. A y = (rank` x)} -> y e. On)
35	27, 34	mpgbir 1334	. . . . . . . 8 |- {y | E.x e. A y = (rank` x)} C_ On
36		onint 3876	. . . . . . . 8 |- (({y | E.x e. A y = (rank` x)} C_ On /\ {y | E.x e. A y = (rank` x)} =/= (/)) -> |^|{y | E.x e. A y = (rank` x)} e. {y | E.x e. A y = (rank` x)})
37	35, 36	mpan 759	. . . . . . 7 |- ({y | E.x e. A y = (rank` x)} =/= (/) -> |^|{y | E.x e. A y = (rank` x)} e. {y | E.x e. A y = (rank` x)})
38	11	dfiin2 3287	. . . . . . 7 |- |^|_x e. A (rank` x) = |^|{y | E.x e. A y = (rank` x)}
39	37, 38	syl5eqel 1975	. . . . . 6 |- ({y | E.x e. A y = (rank` x)} =/= (/) -> |^|_x e. A (rank` x) e. {y | E.x e. A y = (rank` x)})
40	24, 39	syl 12	. . . . 5 |- (A =/= (/) -> |^|_x e. A (rank` x) e. {y | E.x e. A y = (rank` x)})
41		hbii1 3282	. . . . . . . . 9 |- (y e. |^|_x e. A (rank` x) -> A.x y e. |^|_x e. A (rank` x))
42	41	hbeleq 1997	. . . . . . . 8 |- (y = |^|_x e. A (rank` x) -> A.x y = |^|_x e. A (rank` x))
43		eqeq1 1890	. . . . . . . 8 |- (y = |^|_x e. A (rank` x) -> (y = (rank` x) <-> |^|_x e. A (rank` x) = (rank` x)))
44	42, 43	rexbid 2122	. . . . . . 7 |- (y = |^|_x e. A (rank` x) -> (E.x e. A y = (rank` x) <-> E.x e. A |^|_x e. A (rank` x) = (rank` x)))
45	44	elabg 2405	. . . . . 6 |- (|^|_x e. A (rank` x) e. {y | E.x e. A y = (rank` x)} -> (|^|_x e. A (rank` x) e. {y | E.x e. A y = (rank` x)} <-> E.x e. A |^|_x e. A (rank` x) = (rank` x)))
46	45	ibi 652	. . . . 5 |- (|^|_x e. A (rank` x) e. {y | E.x e. A y = (rank` x)} -> E.x e. A |^|_x e. A (rank` x) = (rank` x))
47		sseq1 2637	. . . . . . . 8 |- (|^|_x e. A (rank` x) = (rank` x) -> (|^|_x e. A (rank` x) C_ (rank` y) <-> (rank` x) C_ (rank` y)))
48		ssid 2634	. . . . . . . . . 10 |- (rank` y) C_ (rank` y)
49		fveq2 4681	. . . . . . . . . . . 12 |- (x = y -> (rank` x) = (rank` y))
50	49	sseq1d 2644	. . . . . . . . . . 11 |- (x = y -> ((rank` x) C_ (rank` y) <-> (rank` y) C_ (rank` y)))
51	50	rcla4ev 2381	. . . . . . . . . 10 |- ((y e. A /\ (rank` y) C_ (rank` y)) -> E.x e. A (rank` x) C_ (rank` y))
52	48, 51	mpan2 760	. . . . . . . . 9 |- (y e. A -> E.x e. A (rank` x) C_ (rank` y))
53		iinss 3304	. . . . . . . . 9 |- (E.x e. A (rank` x) C_ (rank` y) -> |^|_x e. A (rank` x) C_ (rank` y))
54	52, 53	syl 12	. . . . . . . 8 |- (y e. A -> |^|_x e. A (rank` x) C_ (rank` y))
55	47, 54	syl5bi 225	. . . . . . 7 |- (|^|_x e. A (rank` x) = (rank` x) -> (y e. A -> (rank` x) C_ (rank` y)))
56	55	r19.21aiv 2175	. . . . . 6 |- (|^|_x e. A (rank` x) = (rank` x) -> A.y e. A (rank` x) C_ (rank` y))
57	56	reximi 2198	. . . . 5 |- (E.x e. A |^|_x e. A (rank` x) = (rank` x) -> E.x e. A A.y e. A (rank` x) C_ (rank` y))
58	40, 46, 57	3syl 24	. . . 4 |- (A =/= (/) -> E.x e. A A.y e. A (rank` x) C_ (rank` y))
59		rabn0 2893	. . . 4 |- ({x e. A | A.y e. A (rank` x) C_ (rank` y)} =/= (/) <-> E.x e. A A.y e. A (rank` x) C_ (rank` y))
60	58, 59	sylibr 217	. . 3 |- (A =/= (/) -> {x e. A | A.y e. A (rank` x) C_ (rank` y)} =/= (/))
61	60	necon4i 2069	. 2 |- ({x e. A | A.y e. A (rank` x) C_ (rank` y)} = (/) -> A = (/))
62	3, 61	impbii 174	1 |- (A = (/) <-> {x e. A | A.y e. A (rank` x) C_ (rank` y)} = (/))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875  |^|cint 3214  |^|_ciin 3256  Oncon0 3657  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  scott0s 5849  cplem1 5850  karden 5856

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-iin 3258  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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