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Theorem mrieqvd 15544
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 15539 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
63adantr 467 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  (Moore `  X )
)
74adantr 467 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  S  C_  X )
8 simpr 463 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
96, 1, 7, 8mrieqvlemd 15535 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =  ( N `  S
) ) )
109necon3bbid 2661 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =/=  ( N `  S
) ) )
1110ralbidva 2824 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  <->  A. x  e.  S  ( N `  ( S  \  {
x } ) )  =/=  ( N `  S ) ) )
125, 11bitrd 257 1  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
\  { x }
) )  =/=  ( N `  S )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737    \ cdif 3401    C_ wss 3404   {csn 3968   ` cfv 5582  Moorecmre 15488  mrClscmrc 15489  mrIndcmri 15490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-mre 15492  df-mrc 15493  df-mri 15494
This theorem is referenced by: (None)
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