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Theorem mrissmrid 14899
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 14894 . . 3  |-  ( ph  ->  S  C_  X )
74, 6sstrd 3514 . 2  |-  ( ph  ->  T  C_  X )
81, 2, 3, 6ismri2d 14891 . . . 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 3503 . . . . 5  |-  ( ph  ->  ( x  e.  T  ->  x  e.  S ) )
114ssdifd 3640 . . . . . . 7  |-  ( ph  ->  ( T  \  {
x } )  C_  ( S  \  { x } ) )
126ssdifssd 3642 . . . . . . 7  |-  ( ph  ->  ( S  \  {
x } )  C_  X )
133, 1, 11, 12mrcssd 14882 . . . . . 6  |-  ( ph  ->  ( N `  ( T  \  { x }
) )  C_  ( N `  ( S  \  { x } ) ) )
1413ssneld 3506 . . . . 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 2871 . . 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 14892 1  |-  ( ph  ->  T  e.  I )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5588  Moorecmre 14840  mrClscmrc 14841  mrIndcmri 14842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-mre 14844  df-mrc 14845  df-mri 14846
This theorem is referenced by:  mreexexlem2d  14903  acsfiindd  15667
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