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Theorem mrissmrid 15499
Description: In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrissmrid.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mrissmrid.2  |-  N  =  (mrCls `  A )
mrissmrid.3  |-  I  =  (mrInd `  A )
mrissmrid.4  |-  ( ph  ->  S  e.  I )
mrissmrid.5  |-  ( ph  ->  T  C_  S )
Assertion
Ref Expression
mrissmrid  |-  ( ph  ->  T  e.  I )

Proof of Theorem mrissmrid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mrissmrid.2 . 2  |-  N  =  (mrCls `  A )
2 mrissmrid.3 . 2  |-  I  =  (mrInd `  A )
3 mrissmrid.1 . 2  |-  ( ph  ->  A  e.  (Moore `  X ) )
4 mrissmrid.5 . . 3  |-  ( ph  ->  T  C_  S )
5 mrissmrid.4 . . . 4  |-  ( ph  ->  S  e.  I )
62, 3, 5mrissd 15494 . . 3  |-  ( ph  ->  S  C_  X )
74, 6sstrd 3471 . 2  |-  ( ph  ->  T  C_  X )
81, 2, 3, 6ismri2d 15491 . . . 4  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
95, 8mpbid 213 . . 3  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
104sseld 3460 . . . . 5  |-  ( ph  ->  ( x  e.  T  ->  x  e.  S ) )
114ssdifd 3598 . . . . . . 7  |-  ( ph  ->  ( T  \  {
x } )  C_  ( S  \  { x } ) )
126ssdifssd 3600 . . . . . . 7  |-  ( ph  ->  ( S  \  {
x } )  C_  X )
133, 1, 11, 12mrcssd 15482 . . . . . 6  |-  ( ph  ->  ( N `  ( T  \  { x }
) )  C_  ( N `  ( S  \  { x } ) ) )
1413ssneld 3463 . . . . 5  |-  ( ph  ->  ( -.  x  e.  ( N `  ( S  \  { x }
) )  ->  -.  x  e.  ( N `  ( T  \  {
x } ) ) ) )
1510, 14imim12d 77 . . . 4  |-  ( ph  ->  ( ( x  e.  S  ->  -.  x  e.  ( N `  ( S  \  { x }
) ) )  -> 
( x  e.  T  ->  -.  x  e.  ( N `  ( T 
\  { x }
) ) ) ) )
1615ralimdv2 2830 . . 3  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  ->  A. x  e.  T  -.  x  e.  ( N `  ( T  \  { x }
) ) ) )
179, 16mpd 15 . 2  |-  ( ph  ->  A. x  e.  T  -.  x  e.  ( N `  ( T  \  { x } ) ) )
181, 2, 3, 7, 17ismri2dd 15492 1  |-  ( ph  ->  T  e.  I )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1867   A.wral 2773    \ cdif 3430    C_ wss 3433   {csn 3993   ` cfv 5592  Moorecmre 15440  mrClscmrc 15441  mrIndcmri 15442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-mre 15444  df-mrc 15445  df-mri 15446
This theorem is referenced by:  mreexexlem2d  15503  acsfiindd  16375
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