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Theorem mrieqvd 13633
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 13628 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
63adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  (Moore `  X )
)
74adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  S  C_  X )
8 simpr 447 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
96, 1, 7, 8mrieqvlemd 13624 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =  ( N `  S
) ) )
109necon3bbid 2555 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =/=  ( N `  S
) ) )
1110ralbidva 2635 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  <->  A. x  e.  S  ( N `  ( S  \  {
x } ) )  =/=  ( N `  S ) ) )
125, 11bitrd 244 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619    \ cdif 3225    C_ wss 3228   {csn 3716   ` cfv 5334  Moorecmre 13577  mrClscmrc 13578  mrIndcmri 13579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-mre 13581  df-mrc 13582  df-mri 13583
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