Theorem ismri2dd 17569

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 17568	. 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 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  ‘cfv 6500  Moorecmre 17513  mrClscmrc 17514  mrIndcmri 17515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-mre 17517  df-mri 17519

This theorem is referenced by:  mrissmrid  17576  mreexmrid  17578  acsfiindd  18488