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Theorem mrieqvd 13822
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 13817 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
63adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  (Moore `  X )
)
74adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  S  C_  X )
8 simpr 448 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
96, 1, 7, 8mrieqvlemd 13813 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =  ( N `  S
) ) )
109necon3bbid 2605 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =/=  ( N `  S
) ) )
1110ralbidva 2686 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  <->  A. x  e.  S  ( N `  ( S  \  {
x } ) )  =/=  ( N `  S ) ) )
125, 11bitrd 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670    \ cdif 3281    C_ wss 3284   {csn 3778   ` cfv 5417  Moorecmre 13766  mrClscmrc 13767  mrIndcmri 13768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-mre 13770  df-mrc 13771  df-mri 13772
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