MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismri2dad Unicode version

Theorem ismri2dad 13539
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 13538 . . . 4  |-  ( ph  ->  S  C_  X )
62, 3, 4, 5ismri2d 13535 . . 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 3653 . . . . . . 7  |-  ( (
ph  /\  x  =  Y )  ->  { x }  =  { Y } )
1110difeq2d 3294 . . . . . 6  |-  ( (
ph  /\  x  =  Y )  ->  ( S  \  { x }
)  =  ( S 
\  { Y }
) )
1211fveq2d 5529 . . . . 5  |-  ( (
ph  /\  x  =  Y )  ->  ( N `  ( S  \  { x } ) )  =  ( N `
 ( S  \  { Y } ) ) )
139, 12eleq12d 2351 . . . 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 2887 . 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 1623    e. wcel 1684   A.wral 2543    \ cdif 3149   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485  mrIndcmri 13486
This theorem is referenced by:  mrieqv2d  13541  mreexmrid  13545  mreexexlem2d  13547  acsfiindd  14280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488  df-mri 13490
  Copyright terms: Public domain W3C validator