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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.
StepHypRef 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 )
31, 2sylibr 212 . . . 4  |-  ( a  e.  (Moore `  X
)  ->  a  e.  ~P ~P X )
43ssriv 3493 . . 3  |-  (Moore `  X )  C_  ~P ~P X
54a1i 11 . 2  |-  ( X  e.  V  ->  (Moore `  X )  C_  ~P ~P X )
6 ssid 3508 . . . 4  |-  ~P X  C_ 
~P X
76a1i 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 )
1093adant1 1012 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  C_ 
U. ~P X )
11 unipw 4687 . . . . 5  |-  U. ~P X  =  X
1210, 11syl6sseq 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
) )
14133ad2ant1 1015 . . . 4  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  ( |^| a  e.  ~P X 
<-> 
|^| a  C_  X
) )
1512, 14mpbird 232 . . 3  |-  ( ( X  e.  V  /\  a  C_  ~P X  /\  a  =/=  (/) )  ->  |^| a  e.  ~P X )
167, 8, 15ismred 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 )
1918adantl 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 )
)
2120sselda 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 )
2321, 22syl 16 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  b  C_  ~P X
)
2419, 23sstrd 3499 . . . . . . 7  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X ) )  /\  b  e.  a )  ->  |^| a  C_  ~P X )
2524ex 432 . . . . . 6  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
b  e.  a  ->  |^| a  C_  ~P X
) )
2625exlimdv 1729 . . . . 5  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  ( E. b  b  e.  a  ->  |^| a  C_  ~P X ) )
2717, 26syl5bi 217 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
) )  ->  (
a  =/=  (/)  ->  |^| a  C_ 
~P X ) )
28273impia 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 ) )
3029sselda 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 )
3230, 31syl 16 . . . . 5  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  e.  a )  ->  X  e.  b )
3332ralrimiva 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 ) )
35343ad2ant1 1015 . . . 4  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  ( X  e.  |^| a  <->  A. b  e.  a  X  e.  b ) )
3633, 35mpbird 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 )
)
3837sselda 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 )
4140adantl 464 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| a  C_  c )
4239, 41sstrd 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 )
4538, 42, 43, 44syl3anc 1226 . . . . 5  |-  ( ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  /\  c  e.  a )  ->  |^| b  e.  c )
4645ralrimiva 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 ) )
4947, 48sylbi 195 . . . . 5  |-  ( b  =/=  (/)  ->  ( |^| b  e.  |^| a  <->  A. c  e.  a  |^| b  e.  c ) )
50493ad2ant3 1017 . . . 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 232 . . 3  |-  ( ( ( X  e.  V  /\  a  C_  (Moore `  X )  /\  a  =/=  (/) )  /\  b  C_ 
|^| a  /\  b  =/=  (/) )  ->  |^| b  e.  |^| a )
5228, 36, 51ismred 15091 . 2  |-  ( ( X  e.  V  /\  a  C_  (Moore `  X
)  /\  a  =/=  (/) )  ->  |^| a  e.  (Moore `  X )
)
535, 16, 52ismred 15091 1  |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
Colors of variables: wff setvar class
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
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