MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrieqvd Structured version   Unicode version

Theorem mrieqvd 14699
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 14694 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
63adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  (Moore `  X )
)
74adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  S  C_  X )
8 simpr 461 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
96, 1, 7, 8mrieqvlemd 14690 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =  ( N `  S
) ) )
109necon3bbid 2699 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  ( N `  ( S  \  { x } ) )  =/=  ( N `  S
) ) )
1110ralbidva 2844 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  <->  A. x  e.  S  ( N `  ( S  \  {
x } ) )  =/=  ( N `  S ) ) )
125, 11bitrd 253 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 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    \ cdif 3436    C_ wss 3439   {csn 3988   ` cfv 5529  Moorecmre 14643  mrClscmrc 14644  mrIndcmri 14645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-mre 14647  df-mrc 14648  df-mri 14649
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator