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Theorem ismri2 17581
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 = (mrClsβ€˜π΄)
ismri2.2 𝐼 = (mrIndβ€˜π΄)
Assertion
Ref Expression
ismri2 ((𝐴 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ 𝐼 ↔ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑆
Allowed substitution hints:   𝐼(π‘₯)   𝑁(π‘₯)   𝑋(π‘₯)

Proof of Theorem ismri2
StepHypRef Expression
1 ismri2.1 . . 3 𝑁 = (mrClsβ€˜π΄)
2 ismri2.2 . . 3 𝐼 = (mrIndβ€˜π΄)
31, 2ismri 17580 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝐼 ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))))
43baibd 539 1 ((𝐴 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ 𝐼 ↔ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βˆ– cdif 3938   βŠ† wss 3941  {csn 4621  β€˜cfv 6534  Moorecmre 17531  mrClscmrc 17532  mrIndcmri 17533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-mre 17535  df-mri 17537
This theorem is referenced by:  ismri2d  17582  lindsdom  36986  aacllem  48096
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