Theorem mriss 17668

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 2763	. . 3 ⊢ (mrCls‘𝐴) = (mrCls‘𝐴)
2		mriss.1	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
3	1, 2	ismri 17664	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥})))))
4	3	simprbda 502	1 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077   ∖ cdif 3902   ⊆ wss 3905  {csn 4583  ‘cfv 6522  Moorecmre 17611  mrClscmrc 17612  mrIndcmri 17613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6478  df-fun 6524  df-fv 6530  df-mre 17615  df-mri 17617

This theorem is referenced by:  mrissd  17669