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Theorem mrissmrid 13543
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 13538 . . 3  |-  ( ph  ->  S  C_  X )
74, 6sstrd 3189 . 2  |-  ( ph  ->  T  C_  X )
81, 2, 3, 6ismri2d 13535 . . . 4  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
95, 8mpbid 201 . . 3  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
104sseld 3179 . . . . 5  |-  ( ph  ->  ( x  e.  T  ->  x  e.  S ) )
114ssdifd 3312 . . . . . . 7  |-  ( ph  ->  ( T  \  {
x } )  C_  ( S  \  { x } ) )
126ssdifssd 3314 . . . . . . 7  |-  ( ph  ->  ( S  \  {
x } )  C_  X )
133, 1, 11, 12mrcssd 13526 . . . . . 6  |-  ( ph  ->  ( N `  ( T  \  { x }
) )  C_  ( N `  ( S  \  { x } ) ) )
1413ssneld 3182 . . . . 5  |-  ( ph  ->  ( -.  x  e.  ( N `  ( S  \  { x }
) )  ->  -.  x  e.  ( N `  ( T  \  {
x } ) ) ) )
1510, 14imim12d 68 . . . 4  |-  ( ph  ->  ( ( x  e.  S  ->  -.  x  e.  ( N `  ( S  \  { x }
) ) )  -> 
( x  e.  T  ->  -.  x  e.  ( N `  ( T 
\  { x }
) ) ) ) )
1615ralimdv2 2623 . . 3  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  ->  A. x  e.  T  -.  x  e.  ( N `  ( T  \  { x }
) ) ) )
179, 16mpd 14 . 2  |-  ( ph  ->  A. x  e.  T  -.  x  e.  ( N `  ( T  \  { x } ) ) )
181, 2, 3, 7, 17ismri2dd 13536 1  |-  ( ph  ->  T  e.  I )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485  mrIndcmri 13486
This theorem is referenced by:  mreexexlem2d  13547  acsfiindd  14280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489  df-mri 13490
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