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Theorem mremre 13506
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 13494 . . . . 5  |-  ( a  e.  (Moore `  X
)  ->  a  C_  ~P X )
2 vex 2791 . . . . . 6  |-  a  e. 
_V
32elpw 3631 . . . . 5  |-  ( a  e.  ~P ~P X  <->  a 
C_  ~P X )
41, 3sylibr 203 . . . 4  |-  ( a  e.  (Moore `  X
)  ->  a  e.  ~P ~P X )
54ssriv 3184 . . 3  |-  (Moore `  X )  C_  ~P ~P X
65a1i 10 . 2  |-  ( X  e.  V  ->  (Moore `  X )  C_  ~P ~P X )
7 ssid 3197 . . . 4  |-  ~P X  C_ 
~P X
87a1i 10 . . 3  |-  ( X  e.  V  ->  ~P X  C_  ~P X )
9 pwidg 3637 . . 3  |-  ( X  e.  V  ->  X  e.  ~P X )
10 intssuni2 3887 . . . . . 6  |-  ( ( a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_  U. ~P X )
11103adant1 973 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_ 
U. ~P X )
12 unipw 4224 . . . . 5  |-  U. ~P X  =  X
1311, 12syl6sseq 3224 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_  X )
14 elpw2g 4174 . . . . 5  |-  ( X  e.  V  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
15143ad2ant1 976 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
1613, 15mpbird 223 . . 3  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  e.  ~P X )
178, 9, 16ismred 13504 . 2  |-  ( X  e.  V  ->  ~P X  e.  (Moore `  X
) )
18 n0 3464 . . . . 5  |-  ( a  =/=  (/)  <->  E. b  b  e.  a )
19 intss1 3877 . . . . . . . . 9  |-  ( b  e.  a  ->  |^| a  C_  b )
2019adantl 452 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  b
)
21 simpr 447 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  a  C_  (Moore `  X )
)
2221sselda 3180 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  e.  (Moore `  X ) )
23 mresspw 13494 . . . . . . . . 9  |-  ( b  e.  (Moore `  X
)  ->  b  C_  ~P X )
2422, 23syl 15 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  C_  ~P X
)
2520, 24sstrd 3189 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  ~P X )
2625ex 423 . . . . . 6  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
b  e.  a  ->  |^| a  C_  ~P X
) )
2726exlimdv 1664 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  ( E. b  b  e.  a  ->  |^| a  C_  ~P X ) )
2818, 27syl5bi 208 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
a  =/=  (/)  ->  |^| a  C_ 
~P X ) )
29283impia 1148 . . 3  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  C_  ~P X )
30 simp2 956 . . . . . . 7  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  a  C_  (Moore `  X ) )
3130sselda 3180 . . . . . 6  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  b  e.  (Moore `  X )
)
32 mre1cl 13496 . . . . . 6  |-  ( b  e.  (Moore `  X
)  ->  X  e.  b )
3331, 32syl 15 . . . . 5  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  X  e.  b )
3433ralrimiva 2626 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  A. b  e.  a  X  e.  b )
35 elintg 3870 . . . . 5  |-  ( X  e.  V  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
36353ad2ant1 976 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
3734, 36mpbird 223 . . 3  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  X  e.  |^| a )
38 simp12 986 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  a  C_  (Moore `  X )
)
3938sselda 3180 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  c  e.  (Moore `  X )
)
40 simpl2 959 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  C_ 
|^| a )
41 intss1 3877 . . . . . . . 8  |-  ( c  e.  a  ->  |^| a  C_  c )
4241adantl 452 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| a  C_  c )
4340, 42sstrd 3189 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  C_  c )
44 simpl3 960 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  =/=  (/) )
45 mreintcl 13497 . . . . . 6  |-  ( ( c  e.  (Moore `  X )  /\  b  C_  c  /\  b  =/=  (/) )  ->  |^| b  e.  c )
4639, 43, 44, 45syl3anc 1182 . . . . 5  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| b  e.  c )
4746ralrimiva 2626 . . . 4  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  A. c  e.  a  |^| b  e.  c )
48 intex 4167 . . . . . 6  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
49 elintg 3870 . . . . . 6  |-  ( |^| b  e.  _V  ->  (
|^| b  e.  |^| a 
<-> 
A. c  e.  a 
|^| b  e.  c ) )
5048, 49sylbi 187 . . . . 5  |-  ( b  =/=  (/)  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c ) )
51503ad2ant3 978 . . . 4  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c )
)
5247, 51mpbird 223 . . 3  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  |^| b  e.  |^| a )
5329, 37, 52ismred 13504 . 2  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  e.  (Moore `  X )
)
546, 17, 53ismred 13504 1  |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  mreacs  13560  mreclatdemo  16833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
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