MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mremre Unicode version

Theorem mremre 13756
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 13744 . . . . 5  |-  ( a  e.  (Moore `  X
)  ->  a  C_  ~P X )
2 vex 2902 . . . . . 6  |-  a  e. 
_V
32elpw 3748 . . . . 5  |-  ( a  e.  ~P ~P X  <->  a 
C_  ~P X )
41, 3sylibr 204 . . . 4  |-  ( a  e.  (Moore `  X
)  ->  a  e.  ~P ~P X )
54ssriv 3295 . . 3  |-  (Moore `  X )  C_  ~P ~P X
65a1i 11 . 2  |-  ( X  e.  V  ->  (Moore `  X )  C_  ~P ~P X )
7 ssid 3310 . . . 4  |-  ~P X  C_ 
~P X
87a1i 11 . . 3  |-  ( X  e.  V  ->  ~P X  C_  ~P X )
9 pwidg 3754 . . 3  |-  ( X  e.  V  ->  X  e.  ~P X )
10 intssuni2 4017 . . . . . 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 4355 . . . . 5  |-  U. ~P X  =  X
1311, 12syl6sseq 3337 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_  X )
14 elpw2g 4304 . . . . 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 13754 . 2  |-  ( X  e.  V  ->  ~P X  e.  (Moore `  X
) )
18 n0 3580 . . . . 5  |-  ( a  =/=  (/)  <->  E. b  b  e.  a )
19 intss1 4007 . . . . . . . . 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 3291 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  e.  (Moore `  X ) )
23 mresspw 13744 . . . . . . . . 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 3301 . . . . . . 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 1643 . . . . 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 3291 . . . . . 6  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  b  e.  (Moore `  X )
)
32 mre1cl 13746 . . . . . 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 2732 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  A. b  e.  a  X  e.  b )
35 elintg 4000 . . . . 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 3291 . . . . . 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 4007 . . . . . . . 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 3301 . . . . . 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 13747 . . . . . 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 2732 . . . 4  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  A. c  e.  a  |^| b  e.  c )
48 intex 4297 . . . . . 6  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
49 elintg 4000 . . . . . 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 13754 . 2  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  e.  (Moore `  X )
)
546, 17, 53ismred 13754 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 1547    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   ` cfv 5394  Moorecmre 13734
This theorem is referenced by:  mreacs  13810  mreclatdemo  17083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-mre 13738
  Copyright terms: Public domain W3C validator