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Theorem mrissmrid 14591
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 14586 . . 3  |-  ( ph  ->  S  C_  X )
74, 6sstrd 3378 . 2  |-  ( ph  ->  T  C_  X )
81, 2, 3, 6ismri2d 14583 . . . 4  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
95, 8mpbid 210 . . 3  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
104sseld 3367 . . . . 5  |-  ( ph  ->  ( x  e.  T  ->  x  e.  S ) )
114ssdifd 3504 . . . . . . 7  |-  ( ph  ->  ( T  \  {
x } )  C_  ( S  \  { x } ) )
126ssdifssd 3506 . . . . . . 7  |-  ( ph  ->  ( S  \  {
x } )  C_  X )
133, 1, 11, 12mrcssd 14574 . . . . . 6  |-  ( ph  ->  ( N `  ( T  \  { x }
) )  C_  ( N `  ( S  \  { x } ) ) )
1413ssneld 3370 . . . . 5  |-  ( ph  ->  ( -.  x  e.  ( N `  ( S  \  { x }
) )  ->  -.  x  e.  ( N `  ( T  \  {
x } ) ) ) )
1510, 14imim12d 74 . . . 4  |-  ( ph  ->  ( ( x  e.  S  ->  -.  x  e.  ( N `  ( S  \  { x }
) ) )  -> 
( x  e.  T  ->  -.  x  e.  ( N `  ( T 
\  { x }
) ) ) ) )
1615ralimdv2 2808 . . 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 14584 1  |-  ( ph  ->  T  e.  I )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2727    \ cdif 3337    C_ wss 3340   {csn 3889   ` cfv 5430  Moorecmre 14532  mrClscmrc 14533  mrIndcmri 14534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-mre 14536  df-mrc 14537  df-mri 14538
This theorem is referenced by:  mreexexlem2d  14595  acsfiindd  15359
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