Theorem ismri2 17341

Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ismri2.1	⊢ 𝑁 = (mrCls‘𝐴)
ismri2.2	⊢ 𝐼 = (mrInd‘𝐴)

Assertion

Ref	Expression
ismri2	⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆

Allowed substitution hints:   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2

Step	Hyp	Ref	Expression
1		ismri2.1	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
2		ismri2.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
3	1, 2	ismri 17340	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
4	3	baibd 540	1 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3884   ⊆ wss 3887  {csn 4561  ‘cfv 6433  Moorecmre 17291  mrClscmrc 17292  mrIndcmri 17293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-mre 17295  df-mri 17297

This theorem is referenced by:  ismri2d  17342  lindsdom  35771  aacllem  46505