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Theorem mremre 13821
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 13809 . . . . 5  |-  ( a  e.  (Moore `  X
)  ->  a  C_  ~P X )
2 vex 2951 . . . . . 6  |-  a  e. 
_V
32elpw 3797 . . . . 5  |-  ( a  e.  ~P ~P X  <->  a 
C_  ~P X )
41, 3sylibr 204 . . . 4  |-  ( a  e.  (Moore `  X
)  ->  a  e.  ~P ~P X )
54ssriv 3344 . . 3  |-  (Moore `  X )  C_  ~P ~P X
65a1i 11 . 2  |-  ( X  e.  V  ->  (Moore `  X )  C_  ~P ~P X )
7 ssid 3359 . . . 4  |-  ~P X  C_ 
~P X
87a1i 11 . . 3  |-  ( X  e.  V  ->  ~P X  C_  ~P X )
9 pwidg 3803 . . 3  |-  ( X  e.  V  ->  X  e.  ~P X )
10 intssuni2 4067 . . . . . 6  |-  ( ( a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_  U. ~P X )
11103adant1 975 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_ 
U. ~P X )
12 unipw 4406 . . . . 5  |-  U. ~P X  =  X
1311, 12syl6sseq 3386 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_  X )
14 elpw2g 4355 . . . . 5  |-  ( X  e.  V  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
15143ad2ant1 978 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
1613, 15mpbird 224 . . 3  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  e.  ~P X )
178, 9, 16ismred 13819 . 2  |-  ( X  e.  V  ->  ~P X  e.  (Moore `  X
) )
18 n0 3629 . . . . 5  |-  ( a  =/=  (/)  <->  E. b  b  e.  a )
19 intss1 4057 . . . . . . . . 9  |-  ( b  e.  a  ->  |^| a  C_  b )
2019adantl 453 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  b
)
21 simpr 448 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  a  C_  (Moore `  X )
)
2221sselda 3340 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  e.  (Moore `  X ) )
23 mresspw 13809 . . . . . . . . 9  |-  ( b  e.  (Moore `  X
)  ->  b  C_  ~P X )
2422, 23syl 16 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  C_  ~P X
)
2520, 24sstrd 3350 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  ~P X )
2625ex 424 . . . . . 6  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
b  e.  a  ->  |^| a  C_  ~P X
) )
2726exlimdv 1646 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  ( E. b  b  e.  a  ->  |^| a  C_  ~P X ) )
2818, 27syl5bi 209 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
a  =/=  (/)  ->  |^| a  C_ 
~P X ) )
29283impia 1150 . . 3  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  C_  ~P X )
30 simp2 958 . . . . . . 7  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  a  C_  (Moore `  X ) )
3130sselda 3340 . . . . . 6  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  b  e.  (Moore `  X )
)
32 mre1cl 13811 . . . . . 6  |-  ( b  e.  (Moore `  X
)  ->  X  e.  b )
3331, 32syl 16 . . . . 5  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  X  e.  b )
3433ralrimiva 2781 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  A. b  e.  a  X  e.  b )
35 elintg 4050 . . . . 5  |-  ( X  e.  V  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
36353ad2ant1 978 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
3734, 36mpbird 224 . . 3  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  X  e.  |^| a )
38 simp12 988 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  a  C_  (Moore `  X )
)
3938sselda 3340 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  c  e.  (Moore `  X )
)
40 simpl2 961 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  C_ 
|^| a )
41 intss1 4057 . . . . . . . 8  |-  ( c  e.  a  ->  |^| a  C_  c )
4241adantl 453 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| a  C_  c )
4340, 42sstrd 3350 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  C_  c )
44 simpl3 962 . . . . . 6  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  b  =/=  (/) )
45 mreintcl 13812 . . . . . 6  |-  ( ( c  e.  (Moore `  X )  /\  b  C_  c  /\  b  =/=  (/) )  ->  |^| b  e.  c )
4639, 43, 44, 45syl3anc 1184 . . . . 5  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| b  e.  c )
4746ralrimiva 2781 . . . 4  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  A. c  e.  a  |^| b  e.  c )
48 intex 4348 . . . . . 6  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
49 elintg 4050 . . . . . 6  |-  ( |^| b  e.  _V  ->  (
|^| b  e.  |^| a 
<-> 
A. c  e.  a 
|^| b  e.  c ) )
5048, 49sylbi 188 . . . . 5  |-  ( b  =/=  (/)  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c ) )
51503ad2ant3 980 . . . 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 224 . . 3  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  |^| b  e.  |^| a )
5329, 37, 52ismred 13819 . 2  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  e.  (Moore `  X )
)
546, 17, 53ismred 13819 1  |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   ` cfv 5446  Moorecmre 13799
This theorem is referenced by:  mreacs  13875  mreclatdemo  17152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803
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