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Theorem ismri2 17612
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 17611 . 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 1534   ∈ wcel 2099  βˆ€wral 3058   βˆ– cdif 3944   βŠ† wss 3947  {csn 4629  β€˜cfv 6548  Moorecmre 17562  mrClscmrc 17563  mrIndcmri 17564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-mre 17566  df-mri 17568
This theorem is referenced by:  ismri2d  17613  lindsdom  37087  aacllem  48234
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