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Theorem mriss 13852
Description: An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
mriss.1  |-  I  =  (mrInd `  A )
Assertion
Ref Expression
mriss  |-  ( ( A  e.  (Moore `  X )  /\  S  e.  I )  ->  S  C_  X )

Proof of Theorem mriss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3  |-  (mrCls `  A )  =  (mrCls `  A )
2 mriss.1 . . 3  |-  I  =  (mrInd `  A )
31, 2ismri 13848 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( (mrCls `  A
) `  ( S  \  { x } ) ) ) ) )
43simprbda 607 1  |-  ( ( A  e.  (Moore `  X )  /\  S  e.  I )  ->  S  C_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    \ cdif 3309    C_ wss 3312   {csn 3806   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800  mrIndcmri 13801
This theorem is referenced by:  mrissd  13853
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803  df-mri 13805
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