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Theorem mriss 17648
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 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)

Proof of Theorem mriss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . 3 (mrCls‘𝐴) = (mrCls‘𝐴)
2 mriss.1 . . 3 𝐼 = (mrInd‘𝐴)
31, 2ismri 17644 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥})))))
43simprbda 497 1 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  cdif 3944  wss 3947  {csn 4633  cfv 6554  Moorecmre 17595  mrClscmrc 17596  mrIndcmri 17597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fv 6562  df-mre 17599  df-mri 17601
This theorem is referenced by:  mrissd  17649
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