MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismri2dd Structured version   Unicode version

Theorem ismri2dd 15484
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 15483 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbird 235 1  |-  ( ph  ->  S  e.  I )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1867   A.wral 2773    \ cdif 3430    C_ wss 3433   {csn 3993   ` cfv 5592  Moorecmre 15432  mrClscmrc 15433  mrIndcmri 15434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fv 5600  df-mre 15436  df-mri 15438
This theorem is referenced by:  mrissmrid  15491  mreexmrid  15493  acsfiindd  16367
  Copyright terms: Public domain W3C validator