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Theorem mrieqvd 13868
Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrieqvd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mrieqvd.2  |-  N  =  (mrCls `  A )
mrieqvd.3  |-  I  =  (mrInd `  A )
mrieqvd.4  |-  ( ph  ->  S  C_  X )
Assertion
Ref Expression
mrieqvd  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
\  { x }
) )  =/=  ( N `  S )
) )
Distinct variable groups:    x, A    x, S    ph, x
Allowed substitution hints:    I( x)    N( x)    X( x)

Proof of Theorem mrieqvd
StepHypRef Expression
1 mrieqvd.2 . . 3  |-  N  =  (mrCls `  A )
2 mrieqvd.3 . . 3  |-  I  =  (mrInd `  A )
3 mrieqvd.1 . . 3  |-  ( ph  ->  A  e.  (Moore `  X ) )
4 mrieqvd.4 . . 3  |-  ( ph  ->  S  C_  X )
51, 2, 3, 4ismri2d 13863 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
63adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  (Moore `  X )
)
74adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  S  C_  X )
8 simpr 449 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
96, 1, 7, 8mrieqvlemd 13859 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =  ( N `  S
) ) )
109necon3bbid 2637 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =/=  ( N `  S
) ) )
1110ralbidva 2723 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  <->  A. x  e.  S  ( N `  ( S  \  {
x } ) )  =/=  ( N `  S ) ) )
125, 11bitrd 246 1  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
\  { x }
) )  =/=  ( N `  S )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319    C_ wss 3322   {csn 3816   ` cfv 5457  Moorecmre 13812  mrClscmrc 13813  mrIndcmri 13814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-mre 13816  df-mrc 13817  df-mri 13818
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