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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.
StepHypRef 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 )
31, 2sylibr 217 . . . 4  |-  ( a  e.  (Moore `  X
)  ->  a  e.  ~P ~P X )
43ssriv 3422 . . 3  |-  (Moore `  X )  C_  ~P ~P X
54a1i 11 . 2  |-  ( X  e.  V  ->  (Moore `  X )  C_  ~P ~P X )
6 ssid 3437 . . . 4  |-  ~P X  C_ 
~P X
76a1i 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 )
1093adant1 1048 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_ 
U. ~P X )
11 unipw 4650 . . . . 5  |-  U. ~P X  =  X
1210, 11syl6sseq 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
) )
14133ad2ant1 1051 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
1512, 14mpbird 240 . . 3  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  e.  ~P X )
167, 8, 15ismred 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 )
1918adantl 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 )
)
2120sselda 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 )
2321, 22syl 17 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  C_  ~P X
)
2419, 23sstrd 3428 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  ~P X )
2524ex 441 . . . . . 6  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
b  e.  a  ->  |^| a  C_  ~P X
) )
2625exlimdv 1787 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  ( E. b  b  e.  a  ->  |^| a  C_  ~P X ) )
2717, 26syl5bi 225 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
a  =/=  (/)  ->  |^| a  C_ 
~P X ) )
28273impia 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 ) )
3029sselda 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 )
3230, 31syl 17 . . . . 5  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  X  e.  b )
3332ralrimiva 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 ) )
35343ad2ant1 1051 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
3633, 35mpbird 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 )
)
3837sselda 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 )
4140adantl 473 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| a  C_  c )
4239, 41sstrd 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 )
4538, 42, 43, 44syl3anc 1292 . . . . 5  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| b  e.  c )
4645ralrimiva 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 ) )
4947, 48sylbi 200 . . . . 5  |-  ( b  =/=  (/)  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c ) )
50493ad2ant3 1053 . . . 4  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c )
)
5146, 50mpbird 240 . . 3  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  |^| b  e.  |^| a )
5228, 36, 51ismred 15586 . 2  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  e.  (Moore `  X )
)
535, 16, 52ismred 15586 1  |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
Colors of variables: wff setvar class
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
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