Theorem scott0 8749

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, 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). (Contributed by NM, 15-Oct-2003.)

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 3192	. . 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 3955	. . 3 |- { x e. (/) | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/)
3	1, 2	syl6eq 2672	. 2 |- ( A = (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )
4		n0 3931	. . . . . . . 8 |- ( A =/= (/) <-> E. x x e. A )
5		nfre1 3005	. . . . . . . . 9 |- F/ x E. x e. A ( rank ` x ) = ( rank ` x )
6		eqid 2622	. . . . . . . . . 10 |- ( rank ` x ) = ( rank ` x )
7		rspe 3003	. . . . . . . . . 10 |- ( ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) -> E. x e. A ( rank ` x ) = ( rank ` x ) )
8	6, 7	mpan2 707	. . . . . . . . 9 |- ( x e. A -> E. x e. A ( rank ` x ) = ( rank ` x ) )
9	5, 8	exlimi 2086	. . . . . . . 8 |- ( E. x x e. A -> E. x e. A ( rank ` x ) = ( rank ` x ) )
10	4, 9	sylbi 207	. . . . . . 7 |- ( A =/= (/) -> E. x e. A ( rank ` x ) = ( rank ` x ) )
11		fvex 6201	. . . . . . . . . . 11 |- ( rank ` x ) e. _V
12		eqeq1 2626	. . . . . . . . . . . 12 |- ( y = ( rank ` x ) -> ( y = ( rank ` x ) <-> ( rank ` x ) = ( rank ` x ) ) )
13	12	anbi2d 740	. . . . . . . . . . 11 |- ( y = ( rank ` x ) -> ( ( x e. A /\ y = ( rank ` x ) ) <-> ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) ) )
14	11, 13	spcev 3300	. . . . . . . . . 10 |- ( ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) -> E. y ( x e. A /\ y = ( rank ` x ) ) )
15	14	eximi 1762	. . . . . . . . 9 |- ( E. x ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) -> E. x E. y ( x e. A /\ y = ( rank ` x ) ) )
16		excom 2042	. . . . . . . . 9 |- ( E. y E. x ( x e. A /\ y = ( rank ` x ) ) <-> E. x E. y ( x e. A /\ y = ( rank ` x ) ) )
17	15, 16	sylibr 224	. . . . . . . 8 |- ( E. x ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) -> E. y E. x ( x e. A /\ y = ( rank ` x ) ) )
18		df-rex 2918	. . . . . . . 8 |- ( E. x e. A ( rank ` x ) = ( rank ` x ) <-> E. x ( x e. A /\ ( rank ` x ) = ( rank ` x ) ) )
19		df-rex 2918	. . . . . . . . 9 |- ( E. x e. A y = ( rank ` x ) <-> E. x ( x e. A /\ y = ( rank ` x ) ) )
20	19	exbii 1774	. . . . . . . 8 |- ( E. y E. x e. A y = ( rank ` x ) <-> E. y E. x ( x e. A /\ y = ( rank ` x ) ) )
21	17, 18, 20	3imtr4i 281	. . . . . . 7 |- ( E. x e. A ( rank ` x ) = ( rank ` x ) -> E. y E. x e. A y = ( rank ` x ) )
22	10, 21	syl 17	. . . . . 6 |- ( A =/= (/) -> E. y E. x e. A y = ( rank ` x ) )
23		abn0 3954	. . . . . 6 |- ( { y | E. x e. A y = ( rank ` x ) } =/= (/) <-> E. y E. x e. A y = ( rank ` x ) )
24	22, 23	sylibr 224	. . . . 5 |- ( A =/= (/) -> { y | E. x e. A y = ( rank ` x ) } =/= (/) )
25	11	dfiin2 4555	. . . . . 6 |- |^|_ x e. A ( rank ` x ) = |^| { y | E. x e. A y = ( rank ` x ) }
26		rankon 8658	. . . . . . . . . 10 |- ( rank ` x ) e. On
27		eleq1 2689	. . . . . . . . . 10 |- ( y = ( rank ` x ) -> ( y e. On <-> ( rank ` x ) e. On ) )
28	26, 27	mpbiri 248	. . . . . . . . 9 |- ( y = ( rank ` x ) -> y e. On )
29	28	rexlimivw 3029	. . . . . . . 8 |- ( E. x e. A y = ( rank ` x ) -> y e. On )
30	29	abssi 3677	. . . . . . 7 |- { y | E. x e. A y = ( rank ` x ) } C_ On
31		onint 6995	. . . . . . 7 |- ( ( { 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 ) } )
32	30, 31	mpan 706	. . . . . 6 |- ( { 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 ) } )
33	25, 32	syl5eqel 2705	. . . . 5 |- ( { y | E. x e. A y = ( rank ` x ) } =/= (/) -> |^|_ x e. A ( rank ` x ) e. { y | E. x e. A y = ( rank ` x ) } )
34		nfii1 4551	. . . . . . . . 9 |- F/_ x |^|_ x e. A ( rank ` x )
35	34	nfeq2 2780	. . . . . . . 8 |- F/ x y = |^|_ x e. A ( rank ` x )
36		eqeq1 2626	. . . . . . . 8 |- ( y = |^|_ x e. A ( rank ` x ) -> ( y = ( rank ` x ) <-> |^|_ x e. A ( rank ` x ) = ( rank ` x ) ) )
37	35, 36	rexbid 3051	. . . . . . 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 ) ) )
38	37	elabg 3351	. . . . . 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 ) ) )
39	38	ibi 256	. . . . 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 ) )
40		ssid 3624	. . . . . . . . . 10 |- ( rank ` y ) C_ ( rank ` y )
41		fveq2 6191	. . . . . . . . . . . 12 |- ( x = y -> ( rank ` x ) = ( rank ` y ) )
42	41	sseq1d 3632	. . . . . . . . . . 11 |- ( x = y -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` y ) C_ ( rank ` y ) ) )
43	42	rspcev 3309	. . . . . . . . . 10 |- ( ( y e. A /\ ( rank ` y ) C_ ( rank ` y ) ) -> E. x e. A ( rank ` x ) C_ ( rank ` y ) )
44	40, 43	mpan2 707	. . . . . . . . 9 |- ( y e. A -> E. x e. A ( rank ` x ) C_ ( rank ` y ) )
45		iinss 4571	. . . . . . . . 9 |- ( E. x e. A ( rank ` x ) C_ ( rank ` y ) -> |^|_ x e. A ( rank ` x ) C_ ( rank ` y ) )
46	44, 45	syl 17	. . . . . . . 8 |- ( y e. A -> |^|_ x e. A ( rank ` x ) C_ ( rank ` y ) )
47		sseq1 3626	. . . . . . . 8 |- ( |^|_ x e. A ( rank ` x ) = ( rank ` x ) -> ( |^|_ x e. A ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` y ) ) )
48	46, 47	syl5ib 234	. . . . . . 7 |- ( |^|_ x e. A ( rank ` x ) = ( rank ` x ) -> ( y e. A -> ( rank ` x ) C_ ( rank ` y ) ) )
49	48	ralrimiv 2965	. . . . . 6 |- ( |^|_ x e. A ( rank ` x ) = ( rank ` x ) -> A. y e. A ( rank ` x ) C_ ( rank ` y ) )
50	49	reximi 3011	. . . . 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 ) )
51	24, 33, 39, 50	4syl 19	. . . 4 |- ( A =/= (/) -> E. x e. A A. y e. A ( rank ` x ) C_ ( rank ` y ) )
52		rabn0 3958	. . . 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 ) )
53	51, 52	sylibr 224	. . 3 |- ( A =/= (/) -> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } =/= (/) )
54	53	necon4i 2829	. 2 |- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) -> A = (/) )
55	3, 54	impbii 199	1 |- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   |^|cint 4475   |^|_ciin 4521   Oncon0 5723   ` cfv 5888   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  scott0s  8751  cplem1  8752  karden  8758  scott0f  33977