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Theorem ismri2dd 14892
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 14891 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbird 232 1  |-  ( ph  ->  S  e.  I )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5588  Moorecmre 14840  mrClscmrc 14841  mrIndcmri 14842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-mre 14844  df-mri 14846
This theorem is referenced by:  mrissmrid  14899  mreexmrid  14901  acsfiindd  15667
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