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Theorem ismri2dad 13555
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 13554 . . . 4  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 13551 . . 3  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbid 201 . 2  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
8 ismri2dad.5 . . 3  |-  ( ph  ->  Y  e.  S )
9 simpr 447 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  x  =  Y )
109sneqd 3666 . . . . . . 7  |-  ( (
ph  /\  x  =  Y )  ->  { x }  =  { Y } )
1110difeq2d 3307 . . . . . 6  |-  ( (
ph  /\  x  =  Y )  ->  ( S  \  { x }
)  =  ( S 
\  { Y }
) )
1211fveq2d 5545 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  ( N `  ( S  \  { x } ) )  =  ( N `
 ( S  \  { Y } ) ) )
139, 12eleq12d 2364 . . . 4  |-  ( (
ph  /\  x  =  Y )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  Y  e.  ( N `  ( S  \  { Y } ) ) ) )
1413notbid 285 . . 3  |-  ( (
ph  /\  x  =  Y )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) ) )
158, 14rspcdv 2900 . 2  |-  ( ph  ->  ( A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) )  ->  -.  Y  e.  ( N `  ( S  \  { Y } ) ) ) )
167, 15mpd 14 1  |-  ( ph  ->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162   {csn 3653   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501  mrIndcmri 13502
This theorem is referenced by:  mrieqv2d  13557  mreexmrid  13561  mreexexlem2d  13563  acsfiindd  14296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-mre 13504  df-mri 13506
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