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

Theorem ismri2dd 13818
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 13817 . 2  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbird 224 1  |-  ( ph  ->  S  e.  I )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2670    \ cdif 3281    C_ wss 3284   {csn 3778   ` cfv 5417  Moorecmre 13766  mrClscmrc 13767  mrIndcmri 13768
This theorem is referenced by:  mrissmrid  13825  mreexmrid  13827  acsfiindd  14562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fv 5425  df-mre 13770  df-mri 13772
  Copyright terms: Public domain W3C validator