Theorem mremre 15093

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 15081	. . . . 5 |- ( a e. (Moore ` X ) -> a C_ ~P X )
2		selpw 4006	. . . . 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 3493	. . 3 |- (Moore ` X ) C_ ~P ~P X
5	4	a1i 11	. 2 |- ( X e. V -> (Moore ` X ) C_ ~P ~P X )
6		ssid 3508	. . . 4 |- ~P X C_ ~P X
7	6	a1i 11	. . 3 |- ( X e. V -> ~P X C_ ~P X )
8		pwidg 4012	. . 3 |- ( X e. V -> X e. ~P X )
9		intssuni2 4297	. . . . . 6 |- ( ( a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
10	9	3adant1 1012	. . . . 5 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ U. ~P X )
11		unipw 4687	. . . . 5 |- U. ~P X = X
12	10, 11	syl6sseq 3535	. . . 4 |- ( ( X e. V /\ a C_ ~P X /\ a =/= (/) ) -> |^| a C_ X )
13		elpw2g 4600	. . . . 5 |- ( X e. V -> ( |^| a e. ~P X <-> |^| a C_ X ) )
14	13	3ad2ant1 1015	. . . 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 15091	. 2 |- ( X e. V -> ~P X e. (Moore ` X ) )
17		n0 3793	. . . . 5 |- ( a =/= (/) <-> E. b b e. a )
18		intss1 4286	. . . . . . . . 9 |- ( b e. a -> |^| a C_ b )
19	18	adantl 464	. . . . . . . 8 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ b )
20		simpr 459	. . . . . . . . . 10 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> a C_ (Moore ` X ) )
21	20	sselda 3489	. . . . . . . . 9 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> b e. (Moore ` X ) )
22		mresspw 15081	. . . . . . . . 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 3499	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) ) /\ b e. a ) -> |^| a C_ ~P X )
25	24	ex 432	. . . . . 6 |- ( ( X e. V /\ a C_ (Moore ` X ) ) -> ( b e. a -> |^| a C_ ~P X ) )
26	25	exlimdv 1729	. . . . 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 1191	. . 3 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a C_ ~P X )
29		simp2 995	. . . . . . 7 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> a C_ (Moore ` X ) )
30	29	sselda 3489	. . . . . 6 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b e. a ) -> b e. (Moore ` X ) )
31		mre1cl 15083	. . . . . 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 2868	. . . 4 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> A. b e. a X e. b )
34		elintg 4279	. . . . 5 |- ( X e. V -> ( X e. |^| a <-> A. b e. a X e. b ) )
35	34	3ad2ant1 1015	. . . 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 1025	. . . . . . 7 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> a C_ (Moore ` X ) )
38	37	sselda 3489	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> c e. (Moore ` X ) )
39		simpl2 998	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ |^| a )
40		intss1 4286	. . . . . . . 8 |- ( c e. a -> |^| a C_ c )
41	40	adantl 464	. . . . . . 7 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| a C_ c )
42	39, 41	sstrd 3499	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b C_ c )
43		simpl3 999	. . . . . 6 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> b =/= (/) )
44		mreintcl 15084	. . . . . 6 |- ( ( c e. (Moore ` X ) /\ b C_ c /\ b =/= (/) ) -> |^| b e. c )
45	38, 42, 43, 44	syl3anc 1226	. . . . 5 |- ( ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) /\ c e. a ) -> |^| b e. c )
46	45	ralrimiva 2868	. . . 4 |- ( ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) /\ b C_ |^| a /\ b =/= (/) ) -> A. c e. a |^| b e. c )
47		intex 4593	. . . . . 6 |- ( b =/= (/) <-> |^| b e. _V )
48		elintg 4279	. . . . . 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 1017	. . . 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 15091	. 2 |- ( ( X e. V /\ a C_ (Moore ` X ) /\ a =/= (/) ) -> |^| a e. (Moore ` X ) )
53	5, 16, 52	ismred 15091	1 |- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   E.wex 1617   e. wcel 1823   =/= wne 2649   A.wral 2804   _Vcvv 3106   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   ` cfv 5570  Moorecmre 15071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-mre 15075

This theorem is referenced by:  mreacs  15147  mreclatdemoBAD  19764