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Theorem ismri2d 16116
Description: Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2.1 𝑁 = (mrCls‘𝐴)
ismri2.2 𝐼 = (mrInd‘𝐴)
ismri2d.3 (𝜑𝐴 ∈ (Moore‘𝑋))
ismri2d.4 (𝜑𝑆𝑋)
Assertion
Ref Expression
ismri2d (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆
Allowed substitution hints:   𝜑(𝑥)   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2d
StepHypRef Expression
1 ismri2d.3 . 2 (𝜑𝐴 ∈ (Moore‘𝑋))
2 ismri2d.4 . 2 (𝜑𝑆𝑋)
3 ismri2.1 . . 3 𝑁 = (mrCls‘𝐴)
4 ismri2.2 . . 3 𝐼 = (mrInd‘𝐴)
53, 4ismri2 16115 . 2 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
61, 2, 5syl2anc 691 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195   = wceq 1475  wcel 1977  wral 2896  cdif 3537  wss 3540  {csn 4125  cfv 5804  Moorecmre 16065  mrClscmrc 16066  mrIndcmri 16067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069  df-mri 16071
This theorem is referenced by:  ismri2dd  16117  ismri2dad  16120  mrieqvd  16121  mrieqv2d  16122  mrissmrid  16124
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