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Theorem ismri2dad 14573
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 14572 . . . 4  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 14569 . . 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 3887 . . . . . . 7  |-  ( (
ph  /\  x  =  Y )  ->  { x }  =  { Y } )
1110difeq2d 3472 . . . . . 6  |-  ( (
ph  /\  x  =  Y )  ->  ( S  \  { x }
)  =  ( S 
\  { Y }
) )
1211fveq2d 5693 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  ( N `  ( S  \  { x } ) )  =  ( N `
 ( S  \  { Y } ) ) )
139, 12eleq12d 2509 . . . 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 3074 . 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 1369    e. wcel 1756   A.wral 2713    \ cdif 3323   {csn 3875   ` cfv 5416  Moorecmre 14518  mrClscmrc 14519  mrIndcmri 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fv 5424  df-mre 14522  df-mri 14524
This theorem is referenced by:  mrieqv2d  14575  mreexmrid  14579  mreexexlem2d  14581  acsfiindd  15345
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