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Theorem mriss 16906
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 2824 . . 3 (mrCls‘𝐴) = (mrCls‘𝐴)
2 mriss.1 . . 3 𝐼 = (mrInd‘𝐴)
31, 2ismri 16902 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥})))))
43simprbda 502 1 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  cdif 3916  wss 3919  {csn 4550  cfv 6343  Moorecmre 16853  mrClscmrc 16854  mrIndcmri 16855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fv 6351  df-mre 16857  df-mri 16859
This theorem is referenced by:  mrissd  16907
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