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Theorem ismri2dad 14892
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 14891 . . . 4  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 14888 . . 3  |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
71, 6mpbid 210 . 2  |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )
8 ismri2dad.5 . . 3  |-  ( ph  ->  Y  e.  S )
9 simpr 461 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  x  =  Y )
109sneqd 4039 . . . . . . 7  |-  ( (
ph  /\  x  =  Y )  ->  { x }  =  { Y } )
1110difeq2d 3622 . . . . . 6  |-  ( (
ph  /\  x  =  Y )  ->  ( S  \  { x }
)  =  ( S 
\  { Y }
) )
1211fveq2d 5870 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  ( N `  ( S  \  { x } ) )  =  ( N `
 ( S  \  { Y } ) ) )
139, 12eleq12d 2549 . . . 4  |-  ( (
ph  /\  x  =  Y )  ->  (
x  e.  ( N `
 ( S  \  { x } ) )  <->  Y  e.  ( N `  ( S  \  { Y } ) ) ) )
1413notbid 294 . . 3  |-  ( (
ph  /\  x  =  Y )  ->  ( -.  x  e.  ( N `  ( S  \  { x } ) )  <->  -.  Y  e.  ( N `  ( S 
\  { Y }
) ) ) )
158, 14rspcdv 3217 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473   {csn 4027   ` cfv 5588  Moorecmre 14837  mrClscmrc 14838  mrIndcmri 14839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-mre 14841  df-mri 14843
This theorem is referenced by:  mrieqv2d  14894  mreexmrid  14898  mreexexlem2d  14900  acsfiindd  15664
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