Theorem mriss 17654

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) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋)

Proof of Theorem mriss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ (mrCls‘𝐴) = (mrCls‘𝐴)
2		mriss.1	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
3	1, 2	ismri 17650	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥})))))
4	3	simprbda 498	1 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050   ∖ cdif 3930   ⊆ wss 3933  {csn 4608  ‘cfv 6542  Moorecmre 17601  mrClscmrc 17602  mrIndcmri 17603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6495  df-fun 6544  df-fv 6550  df-mre 17605  df-mri 17607

This theorem is referenced by:  mrissd  17655