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Theorem ismri2 16061
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 = (mrCls‘𝐴)
ismri2.2 𝐼 = (mrInd‘𝐴)
Assertion
Ref Expression
ismri2 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆
Allowed substitution hints:   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2
StepHypRef Expression
1 ismri2.1 . . 3 𝑁 = (mrCls‘𝐴)
2 ismri2.2 . . 3 𝐼 = (mrInd‘𝐴)
31, 2ismri 16060 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
43baibd 945 1 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  cdif 3536  wss 3539  {csn 4124  cfv 5790  Moorecmre 16011  mrClscmrc 16012  mrIndcmri 16013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fv 5798  df-mre 16015  df-mri 16017
This theorem is referenced by:  ismri2d  16062  lindsdom  32376  aacllem  42319
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