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Theorem ismri2dd 13746
Description: Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2.1  |-  N  =  (mrCls `  A )
ismri2.2  |-  I  =  (mrInd `  A )
ismri2d.3  |-  ( ph  ->  A  e.  (Moore `  X ) )
ismri2d.4  |-  ( ph  ->  S  C_  X )
ismri2dd.5  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
Assertion
Ref Expression
ismri2dd  |-  ( ph  ->  S  e.  I )
Distinct variable groups:    x, A    x, S
Allowed substitution hints:    ph( x)    I( x)    N( x)    X( x)

Proof of Theorem ismri2dd
StepHypRef Expression
1 ismri2dd.5 . 2  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
2 ismri2.1 . . 3  |-  N  =  (mrCls `  A )
3 ismri2.2 . . 3  |-  I  =  (mrInd `  A )
4 ismri2d.3 . . 3  |-  ( ph  ->  A  e.  (Moore `  X ) )
5 ismri2d.4 . . 3  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 13745 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbird 223 1  |-  ( ph  ->  S  e.  I )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1647    e. wcel 1715   A.wral 2628    \ cdif 3235    C_ wss 3238   {csn 3729   ` cfv 5358  Moorecmre 13694  mrClscmrc 13695  mrIndcmri 13696
This theorem is referenced by:  mrissmrid  13753  mreexmrid  13755  acsfiindd  14490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fv 5366  df-mre 13698  df-mri 13700
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