Theorem mremre 15588

Description: The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mremre	|- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )

Proof of Theorem mremre

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mresspw 15576	. . . . 5 |- ( a e. (Moore ` X ) -> a C_ ~P X )
2		selpw 3949	. . . . 5 |- ( a e. ~P ~P X <-> a C_ ~P X )
3	1, 2	sylibr 217	. . . 4 |- ( a e. (Moore ` X ) -> a e. ~P ~P X )
4	3	ssriv 3422	. . 3 |- (Moore ` X ) C_ ~P ~P X
5	4	a1i 11	. 2 |- ( X e. V -> (Moore ` X ) C_ ~P ~P X )
6		ssid 3437	. . . 4 |- ~P X C_ ~P X
7	6	a1i 11	. . 3 |- ( X e. V -> ~P X C_ ~P X )
8		pwidg 3955	. . 3 |- ( X e. V -> X e. ~P X )
9		intssuni2 4251	. . . . . 6 |- ( ( a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
10	9	3adant1 1048	. . . . 5 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
11		unipw 4650	. . . . 5 |- U. ~P X = X
12	10, 11	syl6sseq 3464	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ X )
13		elpw2g 4564	. . . . 5 |- ( X e. V -> ( |^| a e. ~P X <-> |^| a C_ X ) )
14	13	3ad2ant1 1051	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> ( |^| a e. ~P X <-> |^| a C_ X ) )
15	12, 14	mpbird 240	. . 3 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a e. ~P X )
16	7, 8, 15	ismred 15586	. 2 |- ( X e. V -> ~P X e. (Moore ` X ) )
17		n0 3732	. . . . 5 |- ( a =/= (/) <-> E. b b e. a )
18		intss1 4241	. . . . . . . . 9 |- ( b e. a -> |^| a C_ b )
19	18	adantl 473	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ b )
20		simpr 468	. . . . . . . . . 10 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> a C_ (Moore ` X ) )
21	20	sselda 3418	. . . . . . . . 9 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b e. (Moore ` X ) )
22		mresspw 15576	. . . . . . . . 9 |- ( b e. (Moore ` X ) -> b C_ ~P X )
23	21, 22	syl 17	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b C_ ~P X )
24	19, 23	sstrd 3428	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ ~P X )
25	24	ex 441	. . . . . 6 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( b e. a -> |^| a C_ ~P X ) )
26	25	exlimdv 1787	. . . . 5 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( E. b b e. a -> |^| a C_ ~P X ) )
27	17, 26	syl5bi 225	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( a =/= (/) -> |^| a C_ ~P X ) )
28	27	3impia 1228	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a C_ ~P X )
29		simp2 1031	. . . . . . 7 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> a C_ (Moore ` X ) )
30	29	sselda 3418	. . . . . 6 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> b e. (Moore ` X ) )
31		mre1cl 15578	. . . . . 6 |- ( b e. (Moore ` X ) -> X e. b )
32	30, 31	syl 17	. . . . 5 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> X e. b )
33	32	ralrimiva 2809	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> A. b e. a X e. b )
34		elintg 4234	. . . . 5 |- ( X e. V -> ( X e. |^| a <-> A. b e. a X e. b ) )
35	34	3ad2ant1 1051	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> ( X e. |^| a <-> A. b e. a X e. b ) )
36	33, 35	mpbird 240	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> X e. |^| a )
37		simp12 1061	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> a C_ (Moore ` X ) )
38	37	sselda 3418	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> c e. (Moore ` X ) )
39		simpl2 1034	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ |^| a )
40		intss1 4241	. . . . . . . 8 |- ( c e. a -> |^| a C_ c )
41	40	adantl 473	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| a C_ c )
42	39, 41	sstrd 3428	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ c )
43		simpl3 1035	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b =/= (/) )
44		mreintcl 15579	. . . . . 6 |- ( ( c e. (Moore ` X ) /\ b C_ c /\ b =/= (/) ) -> |^| b e. c )
45	38, 42, 43, 44	syl3anc 1292	. . . . 5 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| b e. c )
46	45	ralrimiva 2809	. . . 4 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> A. c e. a |^| b e. c )
47		intex 4557	. . . . . 6 |- ( b =/= (/) <-> |^| b e. _V )
48		elintg 4234	. . . . . 6 |- ( |^| b e. _V -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
49	47, 48	sylbi 200	. . . . 5 |- ( b =/= (/) -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
50	49	3ad2ant3 1053	. . . 4 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
51	46, 50	mpbird 240	. . 3 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> |^| b e. |^| a )
52	28, 36, 51	ismred 15586	. 2 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a e. (Moore ` X ) )
53	5, 16, 52	ismred 15586	1 |- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   E.wex 1671   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   |^|cint 4226   ` cfv 5589  Moorecmre 15566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-mre 15570

This theorem is referenced by:  mreacs  15642  mreclatdemoBAD  20189