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Theorem mrissmrid 13592
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 13587 . . 3  |-  ( ph  ->  S  C_  X )
74, 6sstrd 3223 . 2  |-  ( ph  ->  T  C_  X )
81, 2, 3, 6ismri2d 13584 . . . 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 3213 . . . . 5  |-  ( ph  ->  ( x  e.  T  ->  x  e.  S ) )
114ssdifd 3346 . . . . . . 7  |-  ( ph  ->  ( T  \  {
x } )  C_  ( S  \  { x } ) )
126ssdifssd 3348 . . . . . . 7  |-  ( ph  ->  ( S  \  {
x } )  C_  X )
133, 1, 11, 12mrcssd 13575 . . . . . 6  |-  ( ph  ->  ( N `  ( T  \  { x }
) )  C_  ( N `  ( S  \  { x } ) ) )
1413ssneld 3216 . . . . 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 2657 . . 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 13585 1  |-  ( ph  ->  T  e.  I )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1633    e. wcel 1701   A.wral 2577    \ cdif 3183    C_ wss 3186   {csn 3674   ` cfv 5292  Moorecmre 13533  mrClscmrc 13534  mrIndcmri 13535
This theorem is referenced by:  mreexexlem2d  13596  acsfiindd  14329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-mre 13537  df-mrc 13538  df-mri 13539
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