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Theorem ismri2dad 13862
Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2dad.1  |-  N  =  (mrCls `  A )
ismri2dad.2  |-  I  =  (mrInd `  A )
ismri2dad.3  |-  ( ph  ->  A  e.  (Moore `  X ) )
ismri2dad.4  |-  ( ph  ->  S  e.  I )
ismri2dad.5  |-  ( ph  ->  Y  e.  S )
Assertion
Ref Expression
ismri2dad  |-  ( ph  ->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) )

Proof of Theorem ismri2dad
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ismri2dad.4 . . 3  |-  ( ph  ->  S  e.  I )
2 ismri2dad.1 . . . 4  |-  N  =  (mrCls `  A )
3 ismri2dad.2 . . . 4  |-  I  =  (mrInd `  A )
4 ismri2dad.3 . . . 4  |-  ( ph  ->  A  e.  (Moore `  X ) )
53, 4, 1mrissd 13861 . . . 4  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 13858 . . 3  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbid 202 . 2  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
8 ismri2dad.5 . . 3  |-  ( ph  ->  Y  e.  S )
9 simpr 448 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  x  =  Y )
109sneqd 3827 . . . . . . 7  |-  ( (
ph  /\  x  =  Y )  ->  { x }  =  { Y } )
1110difeq2d 3465 . . . . . 6  |-  ( (
ph  /\  x  =  Y )  ->  ( S  \  { x }
)  =  ( S 
\  { Y }
) )
1211fveq2d 5732 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  ( N `  ( S  \  { x } ) )  =  ( N `
 ( S  \  { Y } ) ) )
139, 12eleq12d 2504 . . . 4  |-  ( (
ph  /\  x  =  Y )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  Y  e.  ( N `  ( S  \  { Y } ) ) ) )
1413notbid 286 . . 3  |-  ( (
ph  /\  x  =  Y )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) ) )
158, 14rspcdv 3055 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  ->  -.  Y  e.  ( N `  ( S  \  { Y } ) ) ) )
167, 15mpd 15 1  |-  ( ph  ->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    \ cdif 3317   {csn 3814   ` cfv 5454  Moorecmre 13807  mrClscmrc 13808  mrIndcmri 13809
This theorem is referenced by:  mrieqv2d  13864  mreexmrid  13868  mreexexlem2d  13870  acsfiindd  14603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-mre 13811  df-mri 13813
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