Theorem mremre 16264

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 16252	. . . . 5 |- ( a e. (Moore ` X ) -> a C_ ~P X )
2		selpw 4165	. . . . 5 |- ( a e. ~P ~P X <-> a C_ ~P X )
3	1, 2	sylibr 224	. . . 4 |- ( a e. (Moore ` X ) -> a e. ~P ~P X )
4	3	ssriv 3607	. . 3 |- (Moore ` X ) C_ ~P ~P X
5	4	a1i 11	. 2 |- ( X e. V -> (Moore ` X ) C_ ~P ~P X )
6		ssid 3624	. . . 4 |- ~P X C_ ~P X
7	6	a1i 11	. . 3 |- ( X e. V -> ~P X C_ ~P X )
8		pwidg 4173	. . 3 |- ( X e. V -> X e. ~P X )
9		intssuni2 4502	. . . . . 6 |- ( ( a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
10	9	3adant1 1079	. . . . 5 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
11		unipw 4918	. . . . 5 |- U. ~P X = X
12	10, 11	syl6sseq 3651	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ X )
13		elpw2g 4827	. . . . 5 |- ( X e. V -> ( |^| a e. ~P X <-> |^| a C_ X ) )
14	13	3ad2ant1 1082	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> ( |^| a e. ~P X <-> |^| a C_ X ) )
15	12, 14	mpbird 247	. . 3 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a e. ~P X )
16	7, 8, 15	ismred 16262	. 2 |- ( X e. V -> ~P X e. (Moore ` X ) )
17		n0 3931	. . . . 5 |- ( a =/= (/) <-> E. b b e. a )
18		intss1 4492	. . . . . . . . 9 |- ( b e. a -> |^| a C_ b )
19	18	adantl 482	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ b )
20		simpr 477	. . . . . . . . . 10 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> a C_ (Moore ` X ) )
21	20	sselda 3603	. . . . . . . . 9 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b e. (Moore ` X ) )
22		mresspw 16252	. . . . . . . . 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 3613	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ ~P X )
25	24	ex 450	. . . . . 6 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( b e. a -> |^| a C_ ~P X ) )
26	25	exlimdv 1861	. . . . 5 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( E. b b e. a -> |^| a C_ ~P X ) )
27	17, 26	syl5bi 232	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( a =/= (/) -> |^| a C_ ~P X ) )
28	27	3impia 1261	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a C_ ~P X )
29		simp2 1062	. . . . . . 7 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> a C_ (Moore ` X ) )
30	29	sselda 3603	. . . . . 6 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> b e. (Moore ` X ) )
31		mre1cl 16254	. . . . . 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 2966	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> A. b e. a X e. b )
34		elintg 4483	. . . . 5 |- ( X e. V -> ( X e. |^| a <-> A. b e. a X e. b ) )
35	34	3ad2ant1 1082	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> ( X e. |^| a <-> A. b e. a X e. b ) )
36	33, 35	mpbird 247	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> X e. |^| a )
37		simp12 1092	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> a C_ (Moore ` X ) )
38	37	sselda 3603	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> c e. (Moore ` X ) )
39		simpl2 1065	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ |^| a )
40		intss1 4492	. . . . . . . 8 |- ( c e. a -> |^| a C_ c )
41	40	adantl 482	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| a C_ c )
42	39, 41	sstrd 3613	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ c )
43		simpl3 1066	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b =/= (/) )
44		mreintcl 16255	. . . . . 6 |- ( ( c e. (Moore ` X ) /\ b C_ c /\ b =/= (/) ) -> |^| b e. c )
45	38, 42, 43, 44	syl3anc 1326	. . . . 5 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| b e. c )
46	45	ralrimiva 2966	. . . 4 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> A. c e. a |^| b e. c )
47		intex 4820	. . . . . 6 |- ( b =/= (/) <-> |^| b e. _V )
48		elintg 4483	. . . . . 6 |- ( |^| b e. _V -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
49	47, 48	sylbi 207	. . . . 5 |- ( b =/= (/) -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
50	49	3ad2ant3 1084	. . . 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 247	. . 3 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> |^| b e. |^| a )
52	28, 36, 51	ismred 16262	. 2 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a e. (Moore ` X ) )
53	5, 16, 52	ismred 16262	1 |- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ` cfv 5888  Moorecmre 16242

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  mreacs  16319  mreclatdemoBAD  20900