Theorem kur14lem9 31196

Description: Lemma for kur14 31198. Since the set T is closed under closure and complement, it contains the minimal set S as a subset, so S also has at most 1 4 elements. (Indeed S = T, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	|- J e. Top
kur14lem.x	|- X = U. J
kur14lem.k	|- K = ( cls ` J )
kur14lem.i	|- I = ( int ` J )
kur14lem.a	|- A C_ X
kur14lem.b	|- B = ( X \ ( K ` A ) )
kur14lem.c	|- C = ( K ` ( X \ A ) )
kur14lem.d	|- D = ( I ` ( K ` A ) )
kur14lem.t	|- T = ( ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } ) u. ( { ( I ` C ) , ( K ` D ) , ( I ` ( K ` B ) ) } u. { ( K ` ( I ` C ) ) , ( I ` ( K ` ( I ` A ) ) ) } ) )
kur14lem.s	|- S = |^| { x e. ~P ~P X | ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) }

Assertion

Ref	Expression
kur14lem9	|- ( S e. Fin /\ ( # ` S ) <_ ; 1 4 )

Distinct variable groups:   x, A   x, K   x, y, T   x, X, y

Allowed substitution hints:   A( y)   B( x, y)   C( x, y)   D( x, y)   S( x, y)   I( x, y)   J( x, y)   K( y)

Proof of Theorem kur14lem9

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kur14lem.s	. . 3 |- S = |^| { x e. ~P ~P X | ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) }
2		vex 3203	. . . . . 6 |- s e. _V
3	2	elintrab 4488	. . . . 5 |- ( s e. |^| { x e. ~P ~P X | ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) } <-> A. x e. ~P ~P X ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) -> s e. x ) )
4		ssun1 3776	. . . . . . . 8 |- { A , ( X \ A ) , ( K ` A ) } C_ ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } )
5		ssun1 3776	. . . . . . . . 9 |- ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) C_ ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } )
6		ssun1 3776	. . . . . . . . . 10 |- ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } ) C_ ( ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } ) u. ( { ( I ` C ) , ( K ` D ) , ( I ` ( K ` B ) ) } u. { ( K ` ( I ` C ) ) , ( I ` ( K ` ( I ` A ) ) ) } ) )
7		kur14lem.t	. . . . . . . . . 10 |- T = ( ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } ) u. ( { ( I ` C ) , ( K ` D ) , ( I ` ( K ` B ) ) } u. { ( K ` ( I ` C ) ) , ( I ` ( K ` ( I ` A ) ) ) } ) )
8	6, 7	sseqtr4i 3638	. . . . . . . . 9 |- ( ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) u. { ( K ` B ) , D , ( K ` ( I ` A ) ) } ) C_ T
9	5, 8	sstri 3612	. . . . . . . 8 |- ( { A , ( X \ A ) , ( K ` A ) } u. { B , C , ( I ` A ) } ) C_ T
10	4, 9	sstri 3612	. . . . . . 7 |- { A , ( X \ A ) , ( K ` A ) } C_ T
11		kur14lem.j	. . . . . . . . . . 11 |- J e. Top
12		kur14lem.x	. . . . . . . . . . . 12 |- X = U. J
13	12	topopn 20711	. . . . . . . . . . 11 |- ( J e. Top -> X e. J )
14	11, 13	ax-mp 5	. . . . . . . . . 10 |- X e. J
15	14	elexi 3213	. . . . . . . . 9 |- X e. _V
16		kur14lem.a	. . . . . . . . 9 |- A C_ X
17	15, 16	ssexi 4803	. . . . . . . 8 |- A e. _V
18	17	tpid1 4303	. . . . . . 7 |- A e. { A , ( X \ A ) , ( K ` A ) }
19	10, 18	sselii 3600	. . . . . 6 |- A e. T
20		kur14lem.k	. . . . . . . . 9 |- K = ( cls ` J )
21		kur14lem.i	. . . . . . . . 9 |- I = ( int ` J )
22		kur14lem.b	. . . . . . . . 9 |- B = ( X \ ( K ` A ) )
23		kur14lem.c	. . . . . . . . 9 |- C = ( K ` ( X \ A ) )
24		kur14lem.d	. . . . . . . . 9 |- D = ( I ` ( K ` A ) )
25	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem7 31194	. . . . . . . 8 |- ( y e. T -> ( y C_ X /\ { ( X \ y ) , ( K ` y ) } C_ T ) )
26	25	simprd 479	. . . . . . 7 |- ( y e. T -> { ( X \ y ) , ( K ` y ) } C_ T )
27	26	rgen 2922	. . . . . 6 |- A. y e. T { ( X \ y ) , ( K ` y ) } C_ T
28	25	simpld 475	. . . . . . . . . 10 |- ( y e. T -> y C_ X )
29	15	elpw2 4828	. . . . . . . . . 10 |- ( y e. ~P X <-> y C_ X )
30	28, 29	sylibr 224	. . . . . . . . 9 |- ( y e. T -> y e. ~P X )
31	30	ssriv 3607	. . . . . . . 8 |- T C_ ~P X
32	15	pwex 4848	. . . . . . . . 9 |- ~P X e. _V
33	32	elpw2 4828	. . . . . . . 8 |- ( T e. ~P ~P X <-> T C_ ~P X )
34	31, 33	mpbir 221	. . . . . . 7 |- T e. ~P ~P X
35		eleq2 2690	. . . . . . . . . 10 |- ( x = T -> ( A e. x <-> A e. T ) )
36		sseq2 3627	. . . . . . . . . . 11 |- ( x = T -> ( { ( X \ y ) , ( K ` y ) } C_ x <-> { ( X \ y ) , ( K ` y ) } C_ T ) )
37	36	raleqbi1dv 3146	. . . . . . . . . 10 |- ( x = T -> ( A. y e. x { ( X \ y ) , ( K ` y ) } C_ x <-> A. y e. T { ( X \ y ) , ( K ` y ) } C_ T ) )
38	35, 37	anbi12d 747	. . . . . . . . 9 |- ( x = T -> ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) <-> ( A e. T /\ A. y e. T { ( X \ y ) , ( K ` y ) } C_ T ) ) )
39		eleq2 2690	. . . . . . . . 9 |- ( x = T -> ( s e. x <-> s e. T ) )
40	38, 39	imbi12d 334	. . . . . . . 8 |- ( x = T -> ( ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) -> s e. x ) <-> ( ( A e. T /\ A. y e. T { ( X \ y ) , ( K ` y ) } C_ T ) -> s e. T ) ) )
41	40	rspccv 3306	. . . . . . 7 |- ( A. x e. ~P ~P X ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) -> s e. x ) -> ( T e. ~P ~P X -> ( ( A e. T /\ A. y e. T { ( X \ y ) , ( K ` y ) } C_ T ) -> s e. T ) ) )
42	34, 41	mpi 20	. . . . . 6 |- ( A. x e. ~P ~P X ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) -> s e. x ) -> ( ( A e. T /\ A. y e. T { ( X \ y ) , ( K ` y ) } C_ T ) -> s e. T ) )
43	19, 27, 42	mp2ani 714	. . . . 5 |- ( A. x e. ~P ~P X ( ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) -> s e. x ) -> s e. T )
44	3, 43	sylbi 207	. . . 4 |- ( s e. |^| { x e. ~P ~P X | ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) } -> s e. T )
45	44	ssriv 3607	. . 3 |- |^| { x e. ~P ~P X | ( A e. x /\ A. y e. x { ( X \ y ) , ( K ` y ) } C_ x ) } C_ T
46	1, 45	eqsstri 3635	. 2 |- S C_ T
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 31195	. 2 |- ( T e. Fin /\ ( # ` T ) <_ ; 1 4 )
48		1nn0 11308	. . 3 |- 1 e. NN0
49		4nn0 11311	. . 3 |- 4 e. NN0
50	48, 49	deccl 11512	. 2 |- ; 1 4 e. NN0
51	46, 47, 50	hashsslei 13213	1 |- ( S e. Fin /\ ( # ` S ) <_ ; 1 4 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {cpr 4179   {ctp 4181   U.cuni 4436   |^|cint 4475   class class class wbr 4653   ` cfv 5888   Fincfn 7955   1c1 9937   <_ cle 10075   4c4 11072  ;cdc 11493   #chash 13117   Topctop 20698   intcnt 20821   clsccl 20822

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-hash 13118  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  kur14lem10  31197