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Theorem mriss 16649
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 2826 . . 3 (mrCls‘𝐴) = (mrCls‘𝐴)
2 mriss.1 . . 3 𝐼 = (mrInd‘𝐴)
31, 2ismri 16645 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥})))))
43simprbda 494 1 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3118  cdif 3796  wss 3799  {csn 4398  cfv 6124  Moorecmre 16596  mrClscmrc 16597  mrIndcmri 16598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fv 6132  df-mre 16600  df-mri 16602
This theorem is referenced by:  mrissd  16650
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