Theorem mremre 14646

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 14634	. . . . 5 |- ( a e. (Moore ` X ) -> a C_ ~P X )
2		selpw 3967	. . . . 5 |- ( a e. ~P ~P X <-> a C_ ~P X )
3	1, 2	sylibr 212	. . . 4 |- ( a e. (Moore ` X ) -> a e. ~P ~P X )
4	3	ssriv 3460	. . 3 |- (Moore ` X ) C_ ~P ~P X
5	4	a1i 11	. 2 |- ( X e. V -> (Moore ` X ) C_ ~P ~P X )
6		ssid 3475	. . . 4 |- ~P X C_ ~P X
7	6	a1i 11	. . 3 |- ( X e. V -> ~P X C_ ~P X )
8		pwidg 3973	. . 3 |- ( X e. V -> X e. ~P X )
9		intssuni2 4253	. . . . . 6 |- ( ( a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
10	9	3adant1 1006	. . . . 5 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
11		unipw 4642	. . . . 5 |- U. ~P X = X
12	10, 11	syl6sseq 3502	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ X )
13		elpw2g 4555	. . . . 5 |- ( X e. V -> ( |^| a e. ~P X <-> |^| a C_ X ) )
14	13	3ad2ant1 1009	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> ( |^| a e. ~P X <-> |^| a C_ X ) )
15	12, 14	mpbird 232	. . 3 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a e. ~P X )
16	7, 8, 15	ismred 14644	. 2 |- ( X e. V -> ~P X e. (Moore ` X ) )
17		n0 3746	. . . . 5 |- ( a =/= (/) <-> E. b b e. a )
18		intss1 4243	. . . . . . . . 9 |- ( b e. a -> |^| a C_ b )
19	18	adantl 466	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ b )
20		simpr 461	. . . . . . . . . 10 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> a C_ (Moore ` X ) )
21	20	sselda 3456	. . . . . . . . 9 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b e. (Moore ` X ) )
22		mresspw 14634	. . . . . . . . 9 |- ( b e. (Moore ` X ) -> b C_ ~P X )
23	21, 22	syl 16	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b C_ ~P X )
24	19, 23	sstrd 3466	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ ~P X )
25	24	ex 434	. . . . . 6 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( b e. a -> |^| a C_ ~P X ) )
26	25	exlimdv 1691	. . . . 5 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( E. b b e. a -> |^| a C_ ~P X ) )
27	17, 26	syl5bi 217	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( a =/= (/) -> |^| a C_ ~P X ) )
28	27	3impia 1185	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a C_ ~P X )
29		simp2 989	. . . . . . 7 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> a C_ (Moore ` X ) )
30	29	sselda 3456	. . . . . 6 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> b e. (Moore ` X ) )
31		mre1cl 14636	. . . . . 6 |- ( b e. (Moore ` X ) -> X e. b )
32	30, 31	syl 16	. . . . 5 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> X e. b )
33	32	ralrimiva 2822	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> A. b e. a X e. b )
34		elintg 4236	. . . . 5 |- ( X e. V -> ( X e. |^| a <-> A. b e. a X e. b ) )
35	34	3ad2ant1 1009	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> ( X e. |^| a <-> A. b e. a X e. b ) )
36	33, 35	mpbird 232	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> X e. |^| a )
37		simp12 1019	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> a C_ (Moore ` X ) )
38	37	sselda 3456	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> c e. (Moore ` X ) )
39		simpl2 992	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ |^| a )
40		intss1 4243	. . . . . . . 8 |- ( c e. a -> |^| a C_ c )
41	40	adantl 466	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| a C_ c )
42	39, 41	sstrd 3466	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ c )
43		simpl3 993	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b =/= (/) )
44		mreintcl 14637	. . . . . 6 |- ( ( c e. (Moore ` X ) /\ b C_ c /\ b =/= (/) ) -> |^| b e. c )
45	38, 42, 43, 44	syl3anc 1219	. . . . 5 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| b e. c )
46	45	ralrimiva 2822	. . . 4 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> A. c e. a |^| b e. c )
47		intex 4548	. . . . . 6 |- ( b =/= (/) <-> |^| b e. _V )
48		elintg 4236	. . . . . 6 |- ( |^| b e. _V -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
49	47, 48	sylbi 195	. . . . 5 |- ( b =/= (/) -> ( |^| b e. |^| a <-> A. c e. a |^| b e. c ) )
50	49	3ad2ant3 1011	. . . 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 232	. . 3 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> |^| b e. |^| a )
52	28, 36, 51	ismred 14644	. 2 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a e. (Moore ` X ) )
53	5, 16, 52	ismred 14644	1 |- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   E.wex 1587   e. wcel 1758   =/= wne 2644   A.wral 2795   _Vcvv 3070   C_ wss 3428   (/)c0 3737   ~Pcpw 3960   U.cuni 4191   |^|cint 4228   ` cfv 5518  Moorecmre 14624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-mre 14628

This theorem is referenced by:  mreacs  14700  mreclatdemoBAD  18818