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Theorem ismri2d 13813
Description: Criterion for a subset of the base set in a Moore system to be independent. 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 )
Assertion
Ref Expression
ismri2d  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
Distinct variable groups:    x, A    x, S
Allowed substitution hints:    ph( x)    I( x)    N( x)    X( x)

Proof of Theorem ismri2d
StepHypRef Expression
1 ismri2d.3 . 2  |-  ( ph  ->  A  e.  (Moore `  X ) )
2 ismri2d.4 . 2  |-  ( ph  ->  S  C_  X )
3 ismri2.1 . . 3  |-  N  =  (mrCls `  A )
4 ismri2.2 . . 3  |-  I  =  (mrInd `  A )
53, 4ismri2 13812 . 2  |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) )
61, 2, 5syl2anc 643 1  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   A.wral 2666    \ cdif 3277    C_ wss 3280   {csn 3774   ` cfv 5413  Moorecmre 13762  mrClscmrc 13763  mrIndcmri 13764
This theorem is referenced by:  ismri2dd  13814  ismri2dad  13817  mrieqvd  13818  mrieqv2d  13819  mrissmrid  13821
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-mre 13766  df-mri 13768
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