Theorem ismri2dd 17679

Description: Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ismri2.1	⊢ 𝑁 = (mrCls‘𝐴)
ismri2.2	⊢ 𝐼 = (mrInd‘𝐴)
ismri2d.3	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
ismri2d.4	⊢ (𝜑 → 𝑆 ⊆ 𝑋)
ismri2dd.5	⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))

Assertion

Ref	Expression
ismri2dd	⊢ (𝜑 → 𝑆 ∈ 𝐼)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆

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

Proof of Theorem ismri2dd

Step	Hyp	Ref	Expression
1		ismri2dd.5	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2		ismri2.1	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
3		ismri2.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
4		ismri2d.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5		ismri2d.4	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17678	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbird 257	1 ⊢ (𝜑 → 𝑆 ∈ 𝐼)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∖ cdif 3960   ⊆ wss 3963  {csn 4631  ‘cfv 6563  Moorecmre 17627  mrClscmrc 17628  mrIndcmri 17629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-mre 17631  df-mri 17633

This theorem is referenced by:  mrissmrid  17686  mreexmrid  17688  acsfiindd  18611